Discussion Papers of theMax Planck Institute for

Research on Collective Goods2019/14

Incomplete-Information Games in Large Populations with Anonymity

Martin Hellwig

MAX PLANCKSOCIETY

Discussion Papers of the Max Planck Institute for Research on Collective Goods 2019/14

Incomplete-Information Games in Large Populations with Anonymity

Martin Hellwig

November 2019

Max Planck Institute for Research on Collective Goods, Kurt-Schumacher-Str. 10, D-53113 Bonn https://www.coll.mpg.de

Incomplete-Information Games in LargePopulations with Anonymity ∗

Martin F. HellwigMax Planck Institute for Research on Collective GoodsKurt-Schumacher-Str. 10, D - 53113 Bonn, Germany

hellwig@coll.mpg.de

November 14, 2019

Abstract

The paper provides mathematical foundations for modeling strate-gic interdependence with a continuum of agents where uncertainty hasan aggregate component and an agent-specific component and the lat-ter satisfies a conditional law of large numbers. This decompositionof uncertainty is implied by a condition of anonymity in beliefs, underwhich the agent in question considers the other agents’ types to beessentially pairwise exchangeable. If there is also anonymity in payofffunctions, all strategically relevant aspects of beliefs are contained in anagent’s macro beliefs about the cross-section distribution of the otheragents’types. The paper also gives conditions under which a functionassigning macro beliefs to types is compatible with the existence of acommon prior.Key Words: Incomplete-information games, large populations, be-

lief functions, common priors, exchangeability, conditional indepen-dence, conditional exact law of large numbers.JEL: C70, D82, D83.

∗This paper originated in the challenge posed by a referee’s asking for the mathematicalfoundations of the analysis of common priors in Hellwig (2011), now Sections 5.2 and 5.3 ofthis paper. For helpful discussions and advice, I am grateful to Felix Bierbrauer, ChristophEngel, Alia Gizatulina, Christian Hellwig, and especially Yeneng Sun.

1

1 Introduction

This paper develops mathematical foundations for the study of incomplete-information games with the following properties:

• The payoff for any one agent depends only on the agent’s own char-acteristics and actions and on the cross-section distribution of actionsin the population.

• There are many agents, and each agent considers the effect of his ownactions on the cross-section distribution of actions to be negligible.

• Uncertainty can be decomposed into an aggregate component and anagent-specific component, and the latter satisfies an exact law of largenumbers.

Such games are not covered by the standard approach to studying strate-gic interdependence with incomplete information, which considers gameswith finitely many participants where each participant has beliefs aboutevery other participant’s characteristics and actions.1

However, in many situations, people do not care about who does whatamong the other participants. They only care about how many of themengage in any one course of action. This is particularly true for strategic in-teractions involving large masses of people where aggregate outcomes dependonly on aggregate measures of actions. The following examples illustrate thepoint:

Currency attacks and bank runs: In models of currency attacks andbank runs, the payoff to an agent’s choice to attack or to run depends onhow many agents are also choosing to attack or to run. Any one agent istherefore concerned about the fraction of people in the population that havereceived bad signals and are likely to speculate against the currency or runon the bank.2

1This approach was introduced by Harsanyi (1967/8) and Mertens and Zamir (1985).2Whereas the early models of bank runs in Bryant (1980) and Diamond and Dybvig

(1983) assumed homogeneous information, since Morris and Shin (1998), the literatureon currency attacks and bank runs has aasumed that each agent privately observes anoisy signal of the fundamental. Given the observation of this signal, the agent formsexpectations about the value of the fundamental and about the population share of theset of people who will choose to participate in a currency attack or a bank run. If thechances are good that this population share is high enough for the attack to be successful,the agent will also choose to particpate. In addition to Morris and Shin (1998) see C.Hellwig (2002), Rochet and Vives (2004), Goldstein and Pauzner (2005), and Angeletosand Werning (2006).

2

Insider trading and market microstructure: Strategic behaviorin markets with asymmetric information depends on agents’ expectationsabout the relative importance of information trading and liquidity trading.In organized markets in which the identities of traders are not revealed, theseexpectations concern the distribution of characteristics among the potentialtraders.3

Durable-goods monopoly: A monopolistic seller of a durable com-modity does not care about the potential buyers’individual valuations forhis good, only about the cross-section distribution of these valuations in thepopulation and the implications of this distribution for the demand that hefaces under alternative selling strategies.4

Corporate takeover battles: The outcome of a takeover battle for thecontrol of a widely-held corporation depends on the fractions of shareholdersthat accept the different tender offers, or that reject them. In assessing thealternatives, therefore, each shareholder is concerned with the distributionof actions taken by the other shareholders.5

Electoral competition and voting: In voting, the identities of indi-viduals are irrelevant. Only the fractions of the population that vote for oragainst the given alternatives matter. In models of strategic voting, peopleform expectations about the distribution of other people’s votes. This dis-tribution depends on the distribution of other people’s characteristics, i.e.,preference parameters or realizations of information variables, and on howthese characteritics affect their votes.6

Mechanism design and voting for public-good provision: Effi -cient public-good provision hinges on a comparison of aggregate marginalbenefits and the marginal costs of an additional unit of a public good. Ag-gregate marginal benefits are given by the sum of the marginal benefits ofthe different participants. People’s identities do not matter, only the cross-section distribution of characteristics in the population. In a large economy,this distribution is independent of any one agent’s characteristics, and thedistribution of reported characteristics is independent of any one agent’sreport.7

In such applications, the notion that any one agent is too insignificantto affect aggregate outcomes is usually formalized by assuming that there

3See, for example Kyle (1985, 1989).4See, e.g., Gul et al. (1985).5See Grossman and Hart (1980) and the subsequent literature.6See, e.g. Lindbeck and Weibull (1987), Alesina and Tabellini (1990).7See Bierbrauer and Hellwig (2010, 2015).

3

is a continuum of agents. Uncertainty is decomposed into an aggregatecomponent and an agent-specific component, and a law of large numbers isassumed for the latter. The mathematical diffi culties inherent in the notionof a continuum of agent-specific random variables are usually ignored.

This is where the present paper steps in. I develop a Harsanyi (1967/68)type space framework for studying strategic interdependence when there isa continuum of agents, each one of them insignificant, and each agent hasincomplete information about the state of the world. Each agent’s character-istics are summarized in terms of an abstract "type". This type determinesthe agent’s probabilistic beliefs about the other agents’types as well as theparameters that enter the agent’s payoff function in any strategic game.

Within this formalism, I show that the properties listed above can bederived from two conditions of anonymity, one referring to payoff functions,the other to beliefs. Anonymity in payoff functions postulates that the payoffwhich an agent expects to obtain from a given action in a strategic gamedepends only on the agent’s own type and action and on the cross-sectiondistribution of the other agents’actions. Which of the other agents is takingwhich action makes no difference as long as the cross-section distribution ofactions is the same.

If all agents choose the same strategy, i.e. the same mapping from typesto actions, the cross-section distribution of actions itself determined by thecross-section distribution of types and the common strategy. If differentagents choose different strategies, this decomposition is not available, butthen the condition of anonymity in beliefs ensures that an agent’s expecta-tions about the cross-section distribution of actions is always determined byhis expectations about the cross-section distribution of types and the cross-section distribution of strategies. Under this condition, the agent thinksabout the other agents’types as essentially pairwise exchangeable randomvariables.

Exchangeability has very strong consequences. If agent a treats theother agents’types as essentially pairwise exchangeable random variables, aversion of De Finetti’s Theorem implies that the agent considers the otheragents’types to be essentially pairwise conditionally independent and iden-tically distributed. Moreover, if there are many agents, the conditioningvariable can be taken to be the distribution of the other agents’types acrossthe population. In this case, the agent’s beliefs can be summarized bywhat I will call his macro beliefs, namely his probabilistic beliefs about thecross-section distribution of types of the other agents. Conditional on thiscross-section distribution, the agent considers the other agents’types to beindependent and identically distributed with a common conditional proba-

4

bility distribution that actually coincides with the cross-section distribution.With a continuum of agents, the notion of (conditional) independence

of agents’types involves a well-known mathematical conundrum.8 If thereis any agent-specific uncertainty about types at all, independence of thedifferent agents’ types implies that, depending on the choice of σ-algebraon the space of agents, the assignment of types to agents may be nonmea-surable and the notion of "cross-section distribution of types" may not bewell defined. Specifically, if the space of agents’names is the Lebesgue unitinterval, then, with probability one, the assignment of types to agents isnonmeasurable.

Sun (2006) has proposed to deal with this conundrum by having suffi -ciently large σ-algebras on the space of agents and on the product of thespace of agents and the underlying probability space so that the randomvariables of interest are jointly measurable in agents’names and in states ofnature and, moreover, integration over both agents’names and states of na-ture exhibits the Fubini property that the order of integration does not makea difference to the outcome. Subsequent work has refined this approach.9 Inparticular, Qiao et al. (2016) provide a framework for studying a continuumof exchangeable and of conditionally independent random variables. I willfollow their approach and rely heavily on their results.

In adapting Sun’s approach to games of incomplete information with acontinuum of agents, one must bridge the gap between Sun’s representationof randomness in terms of a complete probability space and Harsanyi’s rep-resentation of an agent’s beliefs as probability measures over constellationsof other agents’types. If one thinks about the belief ba(ta) of agent a withtype ta in isolation, this task is straightforward, though cumbersome. If onethinks about beliefs as a result of conditioning on the observation of one’sown type, the task is more diffi cult, because it is not a priori clear that thecondition of completeness of the probability spaces underlying the beliefsba(ta) for different types ta can be met for all types simultaneously. I willshow that this can be done if the beliefs ba(ta) of different types ta of agenta are mutually absolutely continuous. I refer to such a belief function ascoherent.

The above-mentioned condition of anonymity in beliefs is imposed on thebelief ba(ta) of agent a with type ta. It implies that the belief ba(ta) is fullydetermined by the agent’s macro belief b∗a(ta), i.e. his probabilistic expec-

8For early accounts, see Judd (1985) and Feldman and Gilles (1986).9See Sun and Zhang (2009), Podczeck (2010), and Qiao et al. (2016). Hammond and

Sun (2003, 2008) develop a related approach that involves the limits of arbitrarily largefinite samples from the given measure space of agents.

5

tations about the cross-section distribution of the other agents’types, withthe proviso that, given this cross-section distribution, the agent considers theother agents’types to be conditionally independent and identically distrib-uted with a conditional probability distribution equal to the cross-sectiondistribution. Together with anonymity in payoff functions, this condition en-sures that the agent’s expected payoff in a strategic game is fully determinedby the agent’s macro belief and by the agent’s expectations about the cross-section distribution of the other agents’strategies. All strategically relevantfeatures of the belief ba(ta) are thus contained in the associated macro beliefb∗a(ta).

If the belief function is coherent, the condition that ba(ta) satisfy anonymityin beliefs for all ta is equivalent to the condition that, under any prior thatinduces the belief function ba(·) as a regular conditional distribution, thetypes ta′ , a′ 6= a, of the other agents are essentially pairwise exchangeable.By a version of de Finetti’s theorem again, it follows that the agent’s prioruncertainty about the other agents’ types can also be decomposed into amacro component and a micro component, such that the macro componentconcerns the cross-section distribution of types and the micro componentconcerns each agent’s individual type, with a conditional probability dis-tribution that coincides with the cross-section distribution of types in thepopulation.

The last part of the paper provides a characterization of common priorsand gives conditions for a coherent macro belief function to be compatiblewith a common prior. Whereas, trivially, every probability distribution overcross-section distributions of types can be used to specify a common priorwith associated belief and macro belief functions, not every belief functionthat satisfies anonymity in beliefs is compatible with the existence of a com-mon prior. A macro belief function that is what I call strongly coherent willbe shown to admit a common prior if and only if it satisfies a version of theconsistency condition that Harsanyi’s (1967/68) gave for the existence of acommon prior in a certain two-player game.

6

2 An Incomplete Information Model with a Con-tinuum of Agents

2.1 Agents, Types, and Beliefs

Let (A,A, α) be a complete atomless measure space of agents with α(A) = 1.Given this measure space, an incomplete-information model

Ta,Θa, θa, baa∈A (2.1)

is specified as follows. For each a ∈ A, Ta is a space of abstract "types",and Θa is a space of "payoff types", i.e. a space of parameters that maybe relevant for the agent’s payoffs in any strategic game. Without loss ofgenerality, I assume that the spaces Ta and Θa are the same for all agents,i.e. that, for some T and Θ, Ta = T and Θa = Θ for all a ∈ A. I also assumethat T and Θ are complete separable metric spaces.

Further, θa and ba are mappings such that, for any ta ∈ Ta, θa(ta) ∈Θ determines the agent’s payoff function in any strategic game and ba(ta)specifies the agent’s beliefs about other agents’types when his type is ta.

The belief ba(ta) is a probability measure over possible constellationsof the other agents’ abstract types. The set of such constellations mightin principle be identified with the product TA−a , where A−a := A\a isthe set of agents other than a. However, this product space is awkwardto deal with because, for an arbitrary element of TA−a , the assignment oftypes to agents is not generally measurable and the notion of a cross-sectiondistribution of types is not generally well defined.10 Therefore, I assumethat the belief type ba(ta) of agent a with type ta is concentrated on a (verysmall) subset of possible type constellations.

To specify this subset, I assume that, for some complete probabilityspace (Ωa(ta),Fa(ta), Pa(ta)) and some measurable function τa(·, ·|ta) fromΩa ×A−a to T , agent a with abstract type ta assigns probability one to therange Rτa(ta) ⊂ TA−a of the mapping

ω → τ a(ω|ta) := τa(ω, a′|ta))a′∈A−a . (2.2)

If Rτa(ta) is endowed with the σ-algebra Fa(ta) of sets F such that for someF ∈ Fa(ta), ta′a′∈A−a ∈ F if and only there exists ω ∈ F such thatτ a(ω|ta) = ta′a′∈A−a , belief types can be specified as follows.10See Judd (1985), Feldman and Gilles (1986).

7

Belief Types For any a ∈ A and ta ∈ T, the belief type ba(ta) is given as

ba(ta) = Pa(ta) τ a(·|ta)−1 (2.3)

In the preceding paragraph, I have been deliberately vague about themeasurability of the function τa(·, ·|ta). It might seem natural to requiremeasurability with respect to the usual product σ-algebra Fa(ta)⊗A−a.,where A−a := A′\a|A′ ∈ A is the σ-algebra on A−a, . However, as wasshown by Sun (2006) and Hammond and Sun (2008), this requirement maypreclude any nontrivial uncertainty at the level of individuals. FollowingSun (2006) and Qiao et al. (2016), I therefore impose the weaker require-ment that τa(·, ·|ta) be measurable with respect to a rich Fubini extensionof Fa(ta)⊗A−a, rather than Fa(ta)⊗A−a itself. For this purpose I first in-troduce the notion of a Fubini extension and then the notion of richness ofthis extension.

Fubini Extension Given the complete probability spaces (Ω,F , P ), and(I, I, λ), the probability space (Ω × I,W, Q) is a Fubini extensionof the product space (Ω × I,F ⊗ I, P ⊗ λ) if for any real-valued Q-integrable function f on (Ω× I,W ), (i) the sections f(·, i) and f(ω, ·)are integrable, respectively, on (Ω,F , P ) for λ-almost all i ∈ I , andon (I, I, λ) for P -almost all ω ∈ Ω, and (ii) the functions

i→∫

Ωf(ω, i)dP (ω) and ω →

∫If(ω, i)dλ(i) (2.4)

are integrable, respectively, on (I, I, λ) and (Ω,F , P ) with∫Ω×I

f(ω, i)dQ =

∫Ω

[∫If(ω, i)dλ(i)

]dP (ω) =

∫I

[∫Ωf(ω, i)dP (ω)

]dλ(i)

(2.5)

To reflect the fact that the probability space (Ω×I,W,Q) has (Ω,F , P ),and (I, I, λ) as its marginal spaces, as required by the Fubini property, Iwrite W = F I and Q = P λ, so the notation

(Ω× I,F I, P λ)

indicates that I refer to a Fubini extension of the product (Ω × I,F ⊗ I,P ⊗ λ).

8

Richness of the Fubini Extension A Fubini extension (Ω×I,FI, Pλ) of a product probability space (Ω × I,F ⊗ I, P ⊗ λ) is said to berich if there exists a measurable function h from (Ω× I,F I, P λ)to the unit interval such that (i) the random variables h(·, i), i ∈ I,are essentially pairwise independent, i.e., for λ-almost all i1 ∈ I, therandom variables h(·, i1) and h(·, i2) are independent for λ-almost alli2 ∈ I, and, moreover, (ii) for λ-almost every i ∈ I, the randomvariable h(·, i) has a uniform distribution.

The requirement of richness excludes the product space (Ω × I,F ⊗ I,P ⊗ λ). If h is a measurable function from the product space (Ω× I,F ⊗I,P ⊗ λ) to the unit interval and if the random variables h(·, i), i ∈ I, areessentially pairwise independent, then, as shown in Proposition 2.1 of Sun(2006), the random variables h(·, i), i ∈ I, must be essentially trivial, i.e.,for λ-almost all i ∈ I, h(·, i) must be a constant random variable, which isnot compatible with richness.11

Conditions for the existence of a rich Fubini extension are given in Sun(2006), Sun and Zhang (2009), and Podczeck (2010). In particular, Sun(2006) shows that a rich Fubini extension exists if (I, I, λ) is a hyperfiniteLoeb space. Sun and Zhang (2009) show that, whereas a rich Fubini exten-sion fails to exist if I is the unit interval with the Lebesgue σ-algebra, anextended Lebesgue unit interval, with a larger σ-algebra, does permit theconstruction of a rich Fubini extension of the product (Ω× I,F ⊗I, P ⊗λ).

In the present context, I set (I, I, λ) = (A−a,A−a, α−a), where α−a :=α|A−a is the restriction of the measure α to A−a ⊂ A, and assume that thefunction τa(·, ·|ta) in the specification of agent a’s beliefs is measurable withrespect to a rich Fubini extension Fa(ta)A−a of the product Fa(ta)⊗A−a.

2.2 Strategic Games

Given an incomplete-information model T,Θ, θa, baa∈A, a strategic gameof incomplete information is defined by specifying, for each a ∈ A, an actionset Sa and a payoff function ua : Θ×

∏a′∈A

Sa → R. I assume that the actions

sets Sa are the same for all agents, i.e. that, for some S, Sa = S for alla ∈ A, so the domain of the payoff function ua can be written as Θ× SA. Ialso assume that S is a compact metric space.

11Proposition 4 in Hammond and Sun (2008) provides a version of this result withessential pairwise conditional independence.

9

A strategy for agent a is a function σa : T → S that indicates, for eachta ∈ T, the action the agent is to take if his type is ta. The strategy thatagent a chooses will as a rule depend on the agent’s expectations aboutthe other agents’strategy choices. If σa(·, a′) is the strategy that agent aexpects to be followed by agent a′ ∈ A−a, the expected payoff of agent awith abstract type ta from choosing an action sa is given as∫

Rτ

ua(θa(ta), sa, σa(ta′ , a′)a′∈A−a) dba(ta′a′∈A−a |ta). (2.6)

A strategy s∗a of agent a is a best response to the strategies σa(·, a′) fora′ ∈ A−a that the agent expects the other agents to choose if∫

Rτ

ua(θa(ta), s∗a(ta), σa(ta′ , a′)a′∈A−a) dba(ta′a′∈A−a |ta)

≥∫Rτ

ua(θa(ta), sa, σa(ta′ , a′)a′∈A−a) dba(ta′a′∈A−a |ta) (2.7)

for all ta ∈ T and all sa ∈ S. In this best-response condition, the compar-ison between the agent’s payoff expectation under the choice s∗a(ta) andan alternative choice sa depends on the agent’s expectations about theconstellation σa(ta′ , a′)a′∈A−a of the other agents’actions as well as theagent’s own payoff type θa(ta). The agent’s expectations about the constel-lation σa(ta′ , a′)a′∈A−a of the other agents’actions in turn depends on theagents’expectaions about the constellation ta′a′∞A−a and the constella-tion σa(·, a′)a′∞A−a of strategies that the agent expects the other agentsto follow.12

In this very general formulation, the other agents’names may matter. Itmay make a difference to agent a whether a given action s is taken by agenta′ or by agent a′′; in forming his expectations, the agent may also make adifference between the events ta′ = t and the ta′′ = t, for any t. To eliminatethis possibility, I will introduce two conditions of anonymity under whichthe agent does not care whether a given action or a given type pertains toagent a′ or to agent a′′, one condition on payoff functions and one conditionon belief functions.

12This dependence is the reason why I formulate beliefs in terms of probability measuresover constellations ta′a′∈A−a , rather than pairs (a

′, ta′) like Sun (2006) or Qiao et al.(2016). In Sun (2006) and Qiao et al. (2016), the simpler approach works because theyonly consider applications in which the economic variables that are of interest, such asaggregate resource requirements, take the form

∫f(a, ta)dα(a). This functional form,

however, does not allow for the more general kinds of strategic interdependence thatappear in many games.

10

3 Anonymity

3.1 Anonymity in Payoff Functions

It seems most natural to define anonymity by the condition that payoffs areunchanged under any measurable permutation of the other agents’names. Ifthe measure space (A−a,A−a, α−a) is homogeneous, for example, if (A−a,A−a, α−a)is a hyperfinite Loeb space, this definition of anonymity is equivalent to therequirement that other agents’actions affect the payoff

ua(θa(ta), sa, σa(ta′ , a′)a′∈A−a)

of agent a only through their cross-section distributionD(σa(ta′ , a′)a′∈A−a),an element of the spaceM(S) of probability measures on S.13 Because thelatter requirement is more convenient to work with, I define:

Anonymity in Payoff Functions For any a ∈ A, there exists a functionu∗a : Θ× S ×M(S)→ R such that, for all ta ∈ T and all sa ∈ S,

ua(θa(ta), sa, σa(ta′ , a′)a′∈A−a) = u∗a(θa(ta), sa, D(σa(ta′ , a′)a′∈A−a))(3.1)

for all constellations of actions σa(ta′ , a′)a′∈A−a for which the cross-section distribution D(σa(ta′ , a′)) is well defined.

Remark 3.1 If the map

(a′, ta′)→ σa(ta′ , a′) (3.2)

from A−a × T to S is measurable, then, for any ta ∈ T, the cross-sectiondistribution of actions D(σa(ta′ , a′)a′∈A−a) is ba(ta)-almost surely well de-fined.

Proof. By assumption, ba(ta) is concentrated on the range Rτa(ta) of thefunction τ a(·|ta). Therefore it suffi ces to show that, for every ω ∈ Ω, thecross-section distribution D(σa(τa(ω, a′|ta), a′)a′∈A−a) is well defined.

13See Khan and Sun (1999), Section 4. Homogeneity fails to hold if (A,A, α) isthe Lebesgue unit interval. In this case, the requirement that the strategy constella-tion σa(ta′ , a′)a′∈A−a affects ua only through the distribution D(σa(ta′ , a

′)a′∈A−a) isstronger than the requirement of invariance under measurable permutations of names. Be-cause the Lebesgue σ-algebra is based on neighbourhood structures, the set of measurablepermutations of names is too small for equivalence of the two notions of anonymity.

11

Because the function τ a(·|ta) from Ωa(ta)×A−a to T is measurable withrespect to the Fubini σ-algebra Fa(ta)A−a, the assumption that (3.2) ismeasurable implies that, for given ta, the map

(ω, a′)→ ϕa(ω, a′|ta) := σa(τa(ω, a′|ta), a′)

from Ωa(ta) × A−a to S is also measurable with respect to the Fubini σ-algebra Fa(ta)A−a. For every ω ∈ Ωa(ta), therefore, the section ϕa(ω, ·|ta)of the function ϕa(·, ·|ta) is measurable with respect to A−a, and one maywrite

D(σaa′(τa(ω, a′)a′∈A−a) = α−a (ϕa(ω, ·|ta))−1, (3.3)

i.e., D(σa(τa(ω, a′|ta), a′)a′∈A−a) is well defined.

With anonymity in payoff functions, the expected payoff (2.6) takes theform ∫

Rτ

u∗a(θa(ta), sa, D(σa(ta′ , a′)a′∈A−a)) dba(ta′a′∈A−a |ta). (3.4)

The cross-section distribution of actions, D(σa(ta′ , a′)a′∈A−a), dependson the interplay between the constellations σaa′(·)a′∈A−a and ta′a′∈A−aof strategies and of types. If the agent expects that all other agents use thesame (measurable) strategy σa(·, a′) = σa : T → S, this interplay takes avery simple form and we obtain

D(σa(ta′ , a′)a′∈A−a) = D(ta′a′∈A−a) (σa)−1, (3.5)

where D(ta′a′∈A−a) is the cross-section distribution of types. In this case,the agent is only concerned about the distribution of the other agents’typesand does not care about which agent has which type.

However, the assumption that agent a expects all other agents to use thesame strategy is problematic. After all, the other agents’strategy choices areendogenous. With enough symmetry assumptions on the exogenous data,namely the functions u∗a′ , θa′ , ba′ , in equilibrium, their strategy choices mayin fact be symmetric, but that is a very special case.14 I will therefore followan alternative approach and impose an additional anonymity condition onbeliefs.14Even then, it might be the case that a Bayes-Nash equilibrium involves a symmetric

set of asymmetric strategy choice, rather than a single strategy that is chosen by all.

12

3.2 Anonymity in Beliefs

The fundamental idea is that, in forming his beliefs, agent a treats the otheragents symmetrically in the sense that the joint distribution of the types ofany two of them is unaffected if their names are interchanged.

Anonymity in Beliefs For any a ∈ A and ta ∈ T, the measure ba(ta)satisfies anonymity in beliefs if, under this measure, the types ta′ ofagents a′ 6= a are essentially pairwise exchangeable, i.e., there existsa probability measure pa(·|ta) on T 2 such that, for α−a-almost alla1 ∈ A−a, one has

ba(ta1 ∈ B1 ∩ ta2 ∈ B2|ta) = pa(B1 ×B2|ta) = pa(B2 ×B1|ta)

for α−a-almost all a2 ∈ A−a and all Borel sets B1, B2 ⊂ T.

Anonymity in beliefs corresponds to de Finetti’s notion of exchange-ability. For sequences of random variables, de Finetti’s Theorem asserts theequivalence of exchangeability with the property of conditional independencerelative to some underlying σ-algebra. Kingman (1978) provides a lucid ac-count of the argument and shows that the conditioning σ-algebra may beidentified with the algebra generated by the limiting sample distribution ofthe process; moreover, by a conditional law of large numbers, this limitingsample distribution of the process coincides with the conditional probabil-ity distribution of any one of the random variables given the conditioningσ-algebra.

For models with a continuum of random variables, Hammond and Sun(2003, 2008) have shown that the property of essential pairwise exchange-ability is equivalent to the property of essential pairwise conditional inde-pendence relative to some countably generated σ-algebra, with identical con-ditional distributions. Relying on the framework of a Fubini extension, Qiaoet al. (2016) have shown that the conditioning σ-algebra may be identifiedwith the algebra generated by the cross-section distributions of the randomvariables in question; moreover, by a conditional law of large numbers, theconditional probability distribution of any one of the random variables andthe cross-section sample distribution coincide.

In the following, I use these results to explore the implications of anonymityin beliefs for incomplete-information games with a continuum of agents. Akey role is played by the cross-section type distribution D(ta′a′∈A−a). Thisdistribution is an element of the spaceM (T ) of probability measures on T.

13

Upon endowing this space with the topology of weak convergence and theassociated Borel σ-algebra, one obtains:

Remark 3.2 For any ta ∈ T, the map ta′a′∈A−a → D(ta′a′∈A−a) fromtype constellations in Rτa(ta) to cross-section distributions of types, i.e. mea-sures inM (T ) , is measurable.

Proof. Given the definition of Rτa(ta) and the specification of the σ-algebraFa(ta) on Rτa(ta), it suffi ces to show that the map

ω → δ(ω) := D(τa(ω, a′|ta)a′∈A−a) (3.6)

from Ωa(ta) toM (T ) is measurable. Since T is a complete separable met-rice space andM (T ) has the topology of weak convergence, there exists asequence gi of bounded continuous functions from T to the unit intervalsuch that the mapping

∆→∫

gi(t)d∆(t)

∞i=1

defines a homeomorphism betweenM (T ) and a subset of R∞ (Parthasarathy,1967, p. 47). To prove the lemma, it therefore suffi ces so show that, for anyi, the mapping

ω →∫gi(t)dδ(t|ω) (3.7)

from Ω to R is measurable. The definition of δ implies that, for any ω,

δ(ω) = α−a (τa(ω, ·|ta))−1, (3.8)

and therefore, ∫gi(t)dδ(t|ω) =

∫gi(τa(ω, a

′|ta))dα−a(a′).

Measurability of (3.7) is therefore implied by the assumption that τa(·, ·|ta)is measurable with respect to a Fubini extension.

The following proposition adapts Proposition 3 of Qiao et al. (2016) tothe present setting.

Proposition 3.3 Let D ⊂ Fa(ta) be the σ-algebra on Rτa(ta) that is gener-ated by the mapping

ta′a′∈A−a → D(ta′a′∈A−a). (3.9)

14

Anonymity in beliefs is equivalent to the requirement that, for any ta ∈T, under the measure ba(ta), conditionally on D, the types ta′, a′ ∈ A−a,are essentially pairwise independent and identically distributed, i.e., that forα−a-almost all a1 ∈ A−a, the agent considers the types ta1 and ta2 to beconditionally independent given D, for α−a-almost all a2 ∈ A−a.

Moreover, the sample distribution D(ta′a′∈A−a) of types in the popula-tion is ba(ta)-almost surely equal to the common conditional distribution oftypes ta1 in the population, i.e., for α−a-almost all a

1 ∈ A−a, the mapping(3.9) from Rτa(ta) toM(T ) is a regular conditional distribution for ta1 givenD.

Proof. Given the definition of ba(ta), anonymity in beliefs is equivalentto the condition that the random variables τa(·, a′|ta), a′ ∈ A−a, are essen-tially pairwise exchangeable. Moreover, D = τ a(F |ta)|F ∈ D, where Dis the sub-σ-algebra of Fa(ta) that is generated by the mapping (3.6). Theproposition is therefore equivalent to the statement that, with anonymityin beliefs, conditionally on D, the random variables τa(·, a′|ta), a′ ∈ A−a,are essentially pairwise independent and identically distributed with regularconditional probability distribution δ(·). This latter statement is implied byProposition 3 of Qiao et al. (2016).

Proposition 3.3 has two components. One component asserts the equiv-alence of anonymity in beliefs with essential pairwise conditional indepen-dence (with identical conditional distributions). The other component as-serts a conditional law of large numbers, namely under the belief ba(ta),the cross-section distribution of types is almost surely equal to the commonconditional distribution of the other agents’ types given the σ-algebra D,which in turn implies that the sample distributions can be interpreted asconditional probability distributions.

3.3 Expected Payoffs under Anonymity in Beliefs

Proposition 3.3 provides the basis for the following result about the relationbetween the cross-section distributions of actions and of types.

Proposition 3.4 If the map (a′, ta′)→ σaa′(ta′) from A−a× T to S is mea-surable, then, under the assumption of anonymity in beliefs,

D(σa(ta′ , a′)a′∈A−a) =

∫A−a

D(ta′a′∈A−a) σa(·, a′)−1dα−a(a′), (3.10)

ba(ta)-almost surely.

15

Proof. Since ba(ta) is concentrated on the range Rτa(ta) of the functionτ a(·|ta), it suffi ces to show that the equation

D(σa(τa(ω, a′|ta), a′)a′∈A−a) =

∫A−a

δ(ω) σa(·, a′)−1dα−a(a′) (3.11)

holds for Pa(ta)-almost all ω ∈ Ωa(ta), where, for any ω, δ(ω) ∈ M(T )is again given by (3.6). Since S is a complete separable metric space andM (S) has the topology of weak convergence, there exists a sequence hiof continuous functions from S to the unit interval such that the mapping

∆→∫

Shi(s)d∆(s)

∞i=1

defines a homeomorphism betweenM (S) and a subset of R∞ (Parthasarathy,1967, p. 47). Thus it suffi ces to prove that, for Pa(ta)-almost all ω ∈ Ωa(ta),the equation∫A.a

hi(σa(τa(ω, a′|ta), a′))dα−a(a′) =

∫A−a

∫Thi(σa(t, a

′))dδ(t|ω)dα−a(a′)

(3.12)holds for all i. For any i, consider the mapping

(ω, a′)→ ηi(ω, a′|ta) := hi(σa(τa(ω, a

′|ta), a′) (3.13)

from Ωa(ta) × A−a to the unit interval, and note that the left-hand sideof (3.12) is equal to the integral

∫A.a

ηi(ω, a′|ta)dα−a(a′). Because the map

(a′, ta′) → σa(ta′ , a′) from A−a × T to S is measurable, the map (ω, a′) →

ηi(ω, a′|ta) is measurable with respect to the Fubini extension Fa(ta)A−a

of the product σ-algebra Fa(ta)⊗A−a.By the argument in the proof of Proposition 3.3, anonymity in beliefs im-

plies that, conditionally on the σ-algebra D that is generated by the mapping(3.6), the random variables ηi(·, a′|ta) are essentially pairwise independent.By Corollary 2 in Qiao et al. (2016), it follows that∫

A.a

ηi(·, a′|ta)dα−a(a′) =

∫A−a

E[ηi(·, a′|ta)|D

]dα−a(a

′) (3.14)

Pa(ta)-almost surely. The argument in the proof of Proposition 3.3 alsoimplies that, for α−a-almost all a′ ∈ A, δ σa(·, a′)−1 (hi)

−1 is a regularconditional distribution for ηi(·, a′|ta) given D. For α−a-almost all a′ ∈ A,therefore,

E[ηi(·, a′|ta)| δ(ω) = ·

]=

∫hi(σa(t, a

′))d∆(t|) (3.15)

16

for all ∆ ∈M(T ), Pa(ta)-almost surely. From (3.13) - (3.15), it follows that,for any i,∫A.a

hi(σa(τa(ω, a′|ta), a′))dα−a(a′) =

∫A.a

ηi(ω, a′|ta)dα−a(a′)

=

∫A−a

E[ηi(·, a′|ta)|δ(ω)

]dα−a(a

′)

=

∫A−a

∫Thi(σa(t, a

′))dδ(t|ω)dα−a(a′)

for all ω outside a Pa(ta)-null set Ni. Therefore (3.12) holds for all i and allω outside the union ∪iNi. Because a countable union of null sets is still anull set, it follows that (3.12) holds for all i, Pa(ta)-almost surely.

Thus, with anonymity in beliefs, the cross-section distribution of ac-tions of agents other than a can be written as a function of the constellationσa(·, a′)a′∈A−a of strategies of the other agents and the cross-section distri-bution D(ta′a′∈A−a) of types of the other agents. By inspection of (3.10),one sees that, in fact, the strategy constellation σa(·, a′)a′∈A−a enters onlythrough the cross-section distribution

D(σa(·, a′)a′∈A−a) := α−a (σa)−1,

where σa is the mapping that assigns to each agent a′ ∈ A−a the strategyσa(·, a′) that agent a expects agent a′ to follow. Equation (3.10) can thusbe rewritten in the form

D(σaa′(ta′)a′∈A−a) =

∫A−a

D(ta′a′∈A−a) s(·)−1d(α−a (σa)−1)

=

∫A−a

D(ta′a′∈A−a) s(·)−1dΣa(s),

whereΣa := α−a (σa)−1 (3.16)

denotes the distribution of other agents’ strategies that is anticipated byagent a.

With anonymity in beliefs as well as payoffs, the expected payoff of agenta with abstract type ta from choosing an action sa can therefore be writtenas

17

∫Rτ

ua(θa(ta), sa, σa(ta′ , a′)a′∈A−a) dba(ta′a′∈A−a |ta)

=

∫Rτ

u∗a(θa(ta), sa, D(σa(ta′ , a′)a′∈A−a)) dba(ta′a′∈A−a |ta)

=

∫Rτ

u∗a

(θa(ta), sa,

∫A−a

D(ta′a′∈A−a) s(·)−1dΣa(s)

)dba(ta′a′∈A−a |ta).

(3.17)

3.4 Expected Payoffs and Macro Beliefs

In (3.17), the agent’s belief ba(ta) matters only to the extent that it concernsthe cross-section type distribution D(ta′a′∈A−a). It is therefore convenientto replace the formulation of beliefs in terms of type constellations by one interms of type distributions. I will refer to such beliefs over type distributionsas macro beliefs.

Macro Beliefs For any a ∈ A and ta ∈ T, the macro belief of agent a withbelief type ba(ta) is a probability measure b∗a(ta) over the spaceM (T )of cross-section distributions of abstract types that is defined by theequation

b∗a(ta) = ba(ta) D(·)−1. (3.18)

From (3.17), one now obtains the following corollary to Proposition 3.4:

Corollary 3.5 With anonymity in payoff functions and in beliefs, the ex-pected payoff of agent a with abstract type ta from choosing an action satakes the form∫

Rτ

u∗a(θa(ta), sa,

∫A−a

∆ s(·)−1dΣa(s) db∗a(∆|ta), (3.19)

A strategy s∗a of agent a is a best response to the constellation σa = σa(·, a′)a′∈A−a

that the agent expects the other agents to choose if and only if the inequality∫Rτ

u∗a(θa(ta), s∗a(ta),

∫A−a

∆ s(·)−1dΣa(s) db∗a(∆|ta)

≥∫Rτ

u∗a(θa(ta), sa,

∫A−a

∆ s(·)−1dΣa(s) db∗a(∆|ta) (3.20)

holds for all ta ∈ T and all sa ∈ S.

18

Anonymity in payoff functions and anonymity in beliefs jointly can thusbe used to transform the incomplete-information model (2.1) in which agentsform beliefs and expectations as to what are the types and actions of everysingle other agent, into a model in which agents only form beliefs and expec-tations about cross-section distribution of types and cross-section distribu-tions of strategies of the other agents. Whereas the other agents’"names"play a substantive role in the best-response condition (2.7), they do not evenappear in the best-response condition (3.20).

The key to this simplification is provided by Propositions 3.3 and 3.4,showing that anonymity in beliefs permits a simple decomposition of anagent’s probabilistic beliefs into a macro component and a micro component,his beliefs about the cross-section distribution of types and his beliefs abouteach individual’s type, conditional on the cross-section distribution. Themicro component of the agent’s beliefs fully determined by the fact that,conditional on the cross-section type distribution, the other agents’ typesare viewed as essentially pairwise independent and identically distributed,with a conditional probability distribution that is equal to the cross-sectiondistribution.

3.5 The Scope of Macro Uncertainty

I conclude this section with a result showing that, if the Fubini extensionFa(ta) A−a is rich, then under anonymity in beliefs, the macro belief ofagent a with type ta can be any measure onM(T ), i.e., there is no restrictionon macro beliefs. The only restriction on admissible beliefs comes from theprinciple that, for a given cross-section distribution of types, the conditionalprobability distribution of the random variable τa(·, a′) that represents thetype of agent a′ 6= a is equal to the cross-section distribution of types, i.e.that the agent’s beliefs about the other agents’types respect the conditionallaw of large numbers that is implied by exchangeability.

Proposition 3.6 Let β be any probability measure onM(T ). If the Fubiniextension (Ωa(ta)×A−a,Fa(ta)A−a, Pa(ta)α−a) is rich, there exists anFa(ta)A−a-measurable mapping τa(·|ta) from Ωa(ta)×A−a to T such thatthe belief ba(ta) that is given by (2.3) satisfies anonymity in beliefs and theassociated macro belief is b∗a(ta) = β.

Proof. The proof proceeds along similar lines as the proof of Proposition5.3 in Sun (2006). Let β ∈ M(M(T )) by given. By standard arguments,

19

there exists anM(T )-valued random variable δβ on (Ωa(ta),Fa(ta), Pa(ta))such that the probability distribution of δβ is β, i.e., Pa(ta) δ

−1β = β.

By Lemma A.5 of Sun (2006), there also exists a measurable function ffromM(T )× [0, 1] into T such that for any δ ∈M(T ),

` f(δ, ·)−1 = δ (3.21)

where ` is the uniform distribution on [0, 1]. Given this function f and therandom variable δβ, define a mapping τa(·, ·|ta) from Ωa(ta) × A−a into Tsuch that, for any ω ∈ Ωa(ta) and a′ ∈ A−a,

τa(ω, a′|ta) = f(δβ(ω), h(ω, a′)), (3.22)

where h is the function given by the richness of the Fubini extension (Ωa(ta)×A−a,Fa(ta)A−a, Pa(ta) α−a).

The function τa(·, ·) is measurable with respect to the σ-algebra Fa(ta)A−a. In fact, τa(·, ·|ta) is the composition of the measurable function f :M(T )× [0, 1] → T with the function H : Ω× A−a →M(T )× [0, 1] that isgiven by setting

H(ω, a′) = (δβ(ω), h(ω, a′))

for any ω ∈ Ωa(ta) and a′ ∈ A−a. Because the map ω → δβ(ω) is measurablewith respect to F and the map (ω, a′)→ h(ω, a′) is measurable with respectto F A−a, the map (ω, a′) → H(ω, a′|ta) is measurable with respect toF A−a, and so is the map (ω, a′)→ τa(ω, a

′|ta) = f(H(ω, a′)).Let C be the sub-σ-algebra of F that is induced by δ. Because T is a

complete separable metric space,M(T ) is also a complete separable metricspace, and C is countably generated. Because the random variables h(·, a′),a′ ∈ A−a, are essentially pairwise independent, Proposition 3 in Hammondand Sun (2006) implies that they are also essentially pairwise conditionallyindependent given C. As in Remark 1 of Hammond and Sun (2008), itfollows that the random pairs (δβ(·), h(·, a′)), a′ ∈ A−a, are also essentiallypairwise conditionally independent given C and so are the random variablesτa(·, a′|ta) = f(δβ(·), h(·, a′)), a′ ∈ A−a. By Proposition 3.3 above, it followsthat the belief ba(ta) that is given by (2.3) satisfies anonymity in beliefs.

Moreover, because, for α−a-almost every a′ ∈ A−a, the random variableh(·, a′) has the uniform distribution `, (3.21) and (3.22) imply that, for α−a-almost every a′ ∈ A−a, conditional on the event δβ(·) = δ, the probabilitydistribution of τa(·, a′|ta) is almost surely equal to δ. For α−a-almost everya′ ∈ A−a, therefore the function δβ(·) is a regular conditional distributionfor τa(·, a′|ta) given the σ-algebra Cδβ that is generated by δβ. By Corollary

20

2 of Qiao et al. (2016) and the fact that, conditionally on Cδβ , the randomvariables τa(·, a′|ta), a′ ∈ A−a, are essentially pairwise independent, one alsoinfers that

D(τa(ω, a′|ta)a′∈A−a) = δβ(ω),

Pa(ta)-almost surely, and, therefore, that the macro belief b∗a(ta) must coin-cide with the probability distribution β of the random variable δβ.

Except for purposes of notation, the construction in the proof of Propo-sition 3.6 makes no reference to the type ta of agent a. The belief ba(ta) =Pa(ta) τ a(·|ta)−1 that results from this construction is fully determined bythe indicated macro belief β. Moreover, it depends on β only through therandom variable δβ. The following remark states this observation formally.

Remark 3.7 The belief ba(ta) in Proposition 3.6 takes the form ba(ta) =∫M(T ) ba(δ) dβ(δ), where b(·) is a mapping fromM(T ) to the space of beliefs.

4 Coherent Belief Functions, Agent-Specific Pri-ors, and Anonymity in Beliefs

4.1 Coherence of Belief Functions

The game theoretic literature contains two disctinct interpretations of thebelief ba(ta) of agent a with type ta. In one interpretation, this belief is givenas one of the exogenous data of the model. For any a ∈ A, ta ∈ T , the beliefba(ta) was taken as given, without any account of how the probability spaces(Ωa(ta),Fa(ta), Pa(ta)) and mappings τa(·, · |ta) for different ta and differenta relate to each other.

In another interpretation, the belief ba(ta) of agent a with type ta is in-terpreted as the result of agent a’s observing some information and adaptinghis beliefs accordingly. In this approach, the type ta of agent a is seen asthe realization of a random variable ta, and the belief ba(ta) is the result ofconditioning on the event ta = ta.

With this interpretation of belief functions, the question of how the prob-ability spaces (Ωa(ta),Fa(ta), Pa(ta)) and mappings τa(·, · |ta) for differentta and different a relate to each other cannot be avoided. Two distinctquestions arise. First, for any given agent a, under what conditions can thebeliefs ba(ta), ta ∈ T, be interpreted as resulting from the agent’s condition-ing on the observation of ta? Second, if the beliefs ba(ta), ta ∈ T, of all agents

21

a ∈ A can be interpreted as results of conditioning, under what conditionscan the underlying prior be taken to be the same for all agents, i.e. underwhat conditions do the given belief functions admit a common prior? I willaddress these two questions in turn, the first one in the remainder of thissection, the second one in Section 5 below.

If the belief function ba(·) of agent a is to be interpreted as a regularconditional distribution, there must be some underlying probability spacethat is independent of ta. The question is whether such a space can bespecified in a way that is compatible with the assumption that, for each ta ∈T, the probability measure ba(ta) is the distribution of a random variableτ a(·|ta) = τa(·, a′|ta)a′∈A−a that is defined on some complete probabilityspace (Ωa(ta),Fa(ta), Pa(ta)) where τa(·, · |ta) is measurable with respect toa rich Fubini extension of the product Fa(ta)×A−a.

The answer to this question turns out to be positive if the belief functionsatisfies an additional condition of coherence, which ensures that the beliefsba(ta) that are associated with different types ta ∈ T all have the samedomain and the same null sets.

Coherence of Belief Functions The belief function ba(·) is said to becoherent if, for any ta and ta in T, the probability measures ba(ta)and ba(ta) are mutually absolutely continuous, i.e., any set F ∈ TA−asatisfies F ∈ Fa(ta) and ba(F |ta) > 0 if and only if it also satisfiesF ∈ Fa(ta) and ba(F |ta) > 0.

Lemma 4.1 If the belief function ba(·) is coherent, the ranges Rτa(tA) of themappings τa(·, ·|ta) and the σ-algebras Fa(ta) are the same for all ta ∈ T, i.e.there exists a measurable space (Ra, Fa) such that for all ta ∈ T,Rτa(tA) =

Ra ⊂ TA−a and Fa(ta) = Fa.

Proof. Fix some ta ∈ T. I claim that, for any ta ∈ T and any F ∈ Fa(ta), itmust be the case that F ∈ Fa(ta). If ba(F |ta) > 0, the claim follows directlyfrom the assumption that the belief function is coherent. If ba(F |ta) = 0, theclaim follows by observing that Rτa(tA)\F ∈ Fa(ta) and ba(Rτa(tA)\F |ta) =

1, so by the coherence of the belief function ba(·), Rτa(tA)\F ∈ Fa(ta) andtherefore F ∈ Fa(ta). Thus Fa(ta) ⊂ Fa(ta).

Since Rτa(tA) ∈ Fa(ta), for any ta ⊂ T, it follows that Rτa(tA) ∈ Fa(ta)and therefore, that Rτa(tA) ⊂ Rτa(tA).

22

By a completely symmetric argument, it must also be the case that, forany ta ∈ T, Fa(ta) ⊂ Fa(ta) and Rτa(tA) ⊂ Rτa(tA). The lemma followsimmediately.

Proposition 4.2 If the belief function ba(·) is coherent, the measurablespaces (Ωa(ta),Fa(ta)) and the mappings τa(·, ·|ta) can be taken to be thesame for all ta, i.e., for some fixed measurable space (Ωa, Fa) and some fixedmapping τa : Ωa ×A−a → T, there exist probability measures Pa(ta), ta ∈ T,such that (i) for every ta ∈ T, the probability space (Ωa, Fa, Pa(ta)) is com-plete, (ii), for every ta ∈ T, ba(ta) = Pa(ta) (τ a)−1, where, for any ω ∈ Ωa,τ a(ω) := τa(ω, a′)a′∈A−a , and (iii) the mapping τa is measurable withrespect to a Fubini extension of the product σ-algebra Fa ⊗A−a.

Proof. Fix some ta ∈ T and set Ωa = Ωa(ta) and τa(·, ·) = τa(·, ·|ta).Given that the range Rτa(ta) is endowed with the σ-algebra Fa(ta), let Fa ⊂Fa(ta) be the coarsest σ-algebra on Ωa with respect to which the functionω → τ a(ω) = τa(ω, a′)a′∈A−a is measurable. Thus, Fa consists of the setsF taking the form

F = ω ∈ Ωa|τ a(ω) ∈ F (F ) (4.1)

for some set F (F ) ∈ Fa(ta).By Lemma 4.1, the coherence of the belief function ba(·) implies that, for

any ta ∈ T and any F ∈ Fa, the set F (F ) belongs to the σ-algebra Fa(ta)so, under the measure ba(ta), the probability ba(F (F )|ta) is well defined.For any ta ∈ T, one may therefore define a probability measure Pa(ta) on(Ωa,Fa) by setting

Pa(F |ta) = ba(F (F )|ta). (4.2)

Given Pa(ta) and (Ωa,Fa), I further define Fa(ta) and Pa(ta) so that(Ωa, Fa(ta), Pa(ta)) is the completion of (Ωa,Fa, Pa(ta)), i.e. Fa(ta) is thesmallest σ-algebra on Ωa that contains all subsets of Pa(ta)-null sets in Fa,and Pa(ta) is the unique extension of Pa(ta) from Fa to Fa(ta).

By the coherence of the belief function ba(·), the Pa(ta)-null sets in Faare the same for all ta ∈ T. Therefore the class of subsets of Pa(ta)-nullsets in Fa is the same for all ta ∈ T. Therefore, the σ-algebra Fa(ta) thatis generated by Fa and the class of subsets of Pa(ta)-null sets in Fa is thesame for all ta ∈ T and can be written as Fa.

By construction, for any ta ∈ T , the probability space (Ωa, Fa, Pa(ta))is complete and ba(ta) = Pa(ta) (τ a)−1, i.e. (Ωa, Fa, Pa(ta)) has properties(i) and (ii) in the statement of the proposition.

23

To prove that (Ωa, Fa, Pa(ta)) also has property (iii), let Fa(ta) A−abe the Fubini extension of Fa(ta)⊗A−a with respect to which the functionτa(·, ·) = τa(·, ·|ta) is measurable. Let W be the class of sets B ∈ Fa(ta) A−a for which the sections Ba′ = ω ∈ Ωa|(ω, a′) ∈ B, a′ ∈ A−a, belongto Fa. One easily verifies that W is an extension of the product σ-algebraFa ⊗A−a and that W inherits the Fubini property from Fa(ta)A−a, i.e.that W is a Fubini extension of Fa ⊗ A−a. Finally, the measurability ofτa(·, ·) with respect to Fa(ta)A−a and the definition of Fa as the coarsestσ-algebra with respect to which the function ω → τ a(ω) = τa(ω, a′)a′∈A−ais measurable imply that τa(·, ·) is also measurable with respect to W.

The Fubini extension of the product σ-algebra Fa ⊗A−a in Proposition4.2 need not be rich. Since Fa is a coarsening of Fa(ta), the σ-algebraFa A−a is a coarsening of Fa(ta)A−a and need not inherit the richnessproperty of the latter. Indeed, Fa A−a will not be rich if, e.g., ba(ta) isa degenerata measure that assigns all probability to a singleton ta′a′∈A−a .In this case, Fa is merely the trivial algebra ∅,Ωa. However, Fa A−awill be rich if the specification of ba(ta) involves nontrivial agent-specificuncertainty, so that the measurability of the function τa(·, ·|ta) depends inan essential way on the richness of Fa(ta)A−a.

4.2 The Existence of Agent-Specific Priors

An interpretation of the belief function as a regular conditional distributionpresumes that the belief function is measurable. Since I have not introduceda σ-algebra on the range of the belief function, i.e., the space of probabilitymeasures on (Ra, Fa), I use the following definition.

Measurability of Belief Functions The belief function ba(·) is said to bemeasurable if and only if, for any Fa-measurable function g from Ra to[0, 1], the function ta →

∫Rag(·)dba(·|ta) from T to [0, 1] is measurable.

Proposition 4.3 Assume that the belief function ba(·) is coherent and mea-surable. Then there exists a complete probability space (Φ, Φ, Πa) and thereexists a function τ from Φ×A to T such that τ is measurable with respect toa Fubini extension of the product σ-algebra Φ⊗A, and ba(τ(·, a)) is a regularconditional distribution for τ(·, a′)a′∈A−a given τ(·, a).

Proof. Let Φ = Ωa × T and Φ = Fa ⊗ B(T ) where Ωa and Fa are givenby Proposition 4.2 and B(T ) is the Borel σ-algebra on T . Let πa be any

24

probability measure on (T,B(T )) and define a measure Πa(πa) as the uniquemeasure on (Φ,Φ) that satisfies

Πa(F ×B|πa) =

∫Fba(F |ta)dπa(ta) (4.3)

for F ∈ Fa and B ∈ B(T ). Further, let Φ(πa) be the σ-algebra on Φ thatis induced by Φ and the class of subsets of sets in Φ that have Πa(πa)-measure zero, and let Πa(πa) be the extension of Πa(πa) to Φ(πa), i.e.the completion of Πa(πa). Then by construction, (Φ, Φ(πa), Πa(πa)) is acomplete probability space.

Next, let τ : Φ × A → T be such that, for any (ω, ta, a′) ∈ Φ × A =

Ωa × T ×A,τ(ω, t, a) = ta if a′ = a (4.4)

andτ(ω, ta, a

′) = τa(ω, a′) if a′ 6= a (4.5)

where τa(·, ·) is given by Proposition 4.2.To obtain the claimed measurability property of τ , consider the class V

of sets V ⊂ Φ×A = Ωa × T ×A that satisfy the following conditions:

ProjTV ∈ B(T ), (4.6)

ProjΩa×AV ∩ [Ωa × a] ∈ Fa ⊗ a, (4.7)

andProjΩa×AV ∩ [Ωa ×A−a] ∈ W, (4.8)

where W is the Fubini extension of the product σ-algebra Fa ⊗ A−a withrespect to which the function τa(·, ·) in Proposition 4.2 is measurable. Oneeasily verifies that V is a Fubini extension of the product σ-algebra Φ⊗A−a.Measurability of the function τ with respect to V follows from (4.4) - (4.8)and the measurability of the function τa(·, ·) with respect to W.

Finally, (4.3), (4.4), and (4.5) imply that, for πa-almost all ta ∈ T, ba(ta)is a conditional distribution for τ(·, a′)a′∈A−a given τ(·, a).

In the preceding proof, the choice of the distribution πa on (T,B(T )) isarbitrary. This observation yields:

Remark 4.4 The measure Πa in Proposition 4.3 can be chosen to have anymarginal distribution on the factor T in the product Φ = Ωa × T.

25

Propositions 4.2 and 4.3 show that, if the belief function ba(·) of agenta is coherent as well as measurable, it can be interpreted as a result of theagent’s conditioning on the observation of his own type in a model with anagent-specific prior. The requirement of coherence is restrictive, but I do notsee any way to eliminate it without losing the property that the probabilityspaces (Ωa(ta),Fa(ta), Pa(ta)) that underlie the specification of the beliefsba(ta) for ta ∈ T are all complete, or equivalently the property that theprobability spaces (Rτa(ta), Fa(ta), ba(ta)), ta ∈ T, are all complete. Asindicated by Lemma 4.1, the property of coherence ensures that the measuresba(ta), ta ∈ T , all have the same domains and the same null sets. Withoutthis property, say if there are two types ta and ta with Fa(ta) 6= Fa(ta),extending the measure ba(ta) to the union Fa(ta) ∪ Fa(ta) and then takingthe completion would seem to run afoul of the fact any suffi ciently rich spacewill have nonmeasurable sets.

4.3 Agent-Specific Priors and Anonymity in Beliefs

In Section 3.2 above, I introduced anonymity in beliefs as a property of themeasure ba(ta) where ta was taken as given. I now consider the implicationsof this property for an agent-specific prior that induces the belief functionba(·) as a regular conditional distribution.

Proposition 4.5 Assume that the belief function ba(·) is coherent and mea-surable. Let (Φ, Φ, Πa) and τ : Φ×A→ T be such that the probability space(Φ, Φ, Πa) is complete, the mapping τ is measurable with respect to a Fu-bini extension of the σ-algebra Φ⊗A, and ba(τ(·, a)) is a regular conditionaldistribution for τ(·, a′)a′∈A−a given τ(·, a). Then the following statementsare equivalent:

(a) For all ϕ ∈ Φ, ba(τ(ϕ, a)) satisfies anonymity in beliefs.(b) Under the measure Πa, the random variables τ(·, a′), a′ ∈ A−a, are es-

sentially pairwise exchangeable, i.e., there exists a probability measure pa(·)on T 2 such that, for α−a-almost all a1 ∈ A−a,

Πa(τ(·, a1)−1(B1) ∩ τ(·, a2)−1(B2)) = pa(B1 ×B2) = pa(B2 ×B1) (4.9)

for α−a-almost all a2 ∈ A−a and all Borel sets B1, B2 ⊂ T.(c) If D ⊂ Φ is the Borel σ-algebra that is generated be the mapping ϕ→

D(τ(ϕ, a′)a′∈A−a), then conditionally on D, the random variables τ(·, a′),a′ ∈ A−a, are essentially pairwise conditionally independent and identi-cally distributed. Moreover, the sample distribution D(τ(ϕ, a′)a′∈A−a) of

26

types in the population is almost surely equal to the common conditonaldistribution of types in the population, i..e., for α−a-almost all a1 ∈ A−a,D(τ(·, a′)a′∈A−a) is a regular conditional distribution for τ(·, a1) given D.

Proof. I first prove that (a) implies (b). The left-hand side of (4.9) can bewritten in the form

Πa(ϕ|τ(ϕ, a1) ∈ B1 ∩ ϕ|τ(ϕ, a2) ∈ B2)

=

∫ΦχB1(τ(ϕ, a1)) · χB2(τ(ϕ, a2))dΠa(ϕ), (4.10)

where χB1 : T → 0, 1 and χB2 : T → 0, 1 are the characteristic functionsof the sets B1 and B2, i.e., for i = 1, 2, χBi(t

i) = 1 if (t, ti) ∈ Bi andχBi(t

i) = 0 if ti /∈ Bi. It therefore suffi ces to show that, if the measureba(τ(ϕ, a)) satisfies anonymity in beliefs for Πa-almost all ϕ ∈ Φ, then∫

Φg(τ(ϕ, a1), τ(ϕ, a2))dΠa(ϕ) =

∫Φg(τ(ϕ, a2), τ(ϕ, a1))dΠa(ϕ) (4.11)

for all measurable functions g : T 2 → [0, 1].For any measurable functions g : T 2 → [0, 1]. the fact that ba(τ(·, a)) is a

regular conditional distribution for τ(·, a′)a′∈A−a given τ(·, a) implies thatthe left-hand side of (4.11) is equal to∫

Φ

∫Ra

g(ta1 , ta2) db(ta′a′∈A−a |τ(ϕ, a))dΠa(ϕ). (4.12)

By Proposition 3.3, statement (a) implies that (4.12) can be rewritten inthe form∫

Φ

∫M(T )

∫T

∫Tg(ta1 , ta2) dδ(ta1)dδ(ta2)db

∗(δ|τ(ϕ, a))dΠa(ϕ). (4.13)

By Fubini’s theorem, (4.13) in turn is equal to∫Φ

∫M(T )

∫T

∫Tg(ta1 , ta2) dδ(ta2)dδ(ta1)db

∗(δ|τ(ϕ, a))dΠa(ϕ). (4.14)

By a relabeling of indices, finally, (4.14) can be rewritten as∫Φ

∫M(T )

∫T

∫Tg(ta2 , ta1) dδ(ta1)dδ(ta2)db

∗(δ|τ(ϕ, a))dΠa(ϕ), (4.15)

which is easily seen to be equal to the right-hand side of (4.11).

27

Next, if (b) holds, then (c) follows from Proposition 3 of Qiao et al.(2016).

Finally, assume that (c) holds. Let C be the σ-algebra that is generatedby the mapping

ϕ→ (τ(ϕ, a), D(τ(ϕ, a′)a′∈A−a)).

For any a1 ∈ A−a and a2 ∈ A−a, let µa1a2(·) be a regular conditional distri-

bution for the pair (τ(·, a1), τ(·, a2)) given C and let

Φa1a2 := ϕ ∈ Φ| µa1a2(ϕ)) = (D(τ(ϕ, a′)a′∈A−a))2 (4.16)

is the set of states ϕ for which µa1a2(ϕ) is equal to the product distribution

(D(τ(ϕ, a′)a′∈A−a))2. I claim that, for α−a-almost all a1 ∈ A−a, one maytherefore infer that Πa(Φ

a1a2) = 1 for α−a-almost all a2 ∈ A−a.Because T is a complete separable metric space,M(T ) is also a complete

separable metric space. By standard arguments, therefore C is countablygenerated. Trivially also, D ⊂ C ⊂Φ. By Proposition 3 of Hammond and Sun(2006) therefore, (c) implies that, conditionally on C, the random variablesτ(·, a′), a′ ∈ A−a, are essentially pairwise conditionally independent. Bythe argument given in the proof of Proposition 3 in Hammond and Sun(2006), (c) also implies that, conditionally on C, the random variables τ(·, a′),a′ ∈ A−a, are essentially identically distributed and that, for α−a-almost alla′′ ∈ A−a, D(τ(·, a′)a′∈A−a) is a regular conditional distribution for τ(·, a′′)given C. The claim that, for α−a-almost all a1 ∈ A−a, Πa(Φ

a1a2) = 1 forα−a-almost all a2 ∈ A−a follows immediately.

Because ba(τ(·, a)) is a regular conditional distribution for τ(·, a′)a′∈A−agiven τ(·, a), Πa(Φ

a1a2) = 1 implies∫bata′a′∈A−a | µa

1a2(ϕ) = D(ta′a′∈A−a)2|τ(ϕ, a)dΠa(ϕ) = 1. (4.17)

Because µa1a2(·) is measurable with respect to C, there exists a function

µa1a2(·) such that

µa1a2(ϕ) = µa

1a2(τ(ϕ, a), D(τ(ϕ, a′)a′∈A−a)) (4.18)

for Πa-almost all ϕ. Moreover, for Πa-almost all ϕ, (c) implies

D(ta′a′∈A−a) = D(τ(ϕ, a′)a′∈A−a)

28

for ba(τ(ϕ, a))-almost all ta′a′∈A−a . Thus (4.17) implies∫bata′a′∈A−a | µa

1a2(τ(ϕ, a), D(ta′a′∈A−a)) = D(ta′a′∈A−a)2|τ(ϕ, a)dΠa(ϕ) = 1

and therefore

ba

(ta′a′∈A−a | µa

1a2(τ(ϕ, a), D(ta′a′∈A−a)) = (D(ta′a′∈A−a)2|τ(ϕ, a))

= 1

(4.19)for Πa-almost all ϕ. Because the belief function ba(·) is coherent, it followsthat

ba

(ta′a′∈A−a | µa

1a2(ta, D(ta′a′∈A−a)) = (D(ta′a′∈A−a)2|ta)

= 1

(4.20)for all ta ∈ T.

Thus, if (c) holds, then for α−a-almost all a1 ∈ A−a, (4.20) holds forα−a-almost all a2 ∈ A−a, for all ta ∈ T. By Proposition 3.3, it follows that,for all ta ∈ T, ba(ta) satisfies anonymity in beliefs.

Thus at the level of priors, anonymity in beliefs translates into a condi-tion of essential pairwise exchangeability and the accompanying conditionallaw of large numbers.

Proposition 4.5 assumes, and the proof uses, the property of coherenceof the belief function ba(·). However, this property is less important herethan in Proposition 4.3 above. In the proof that (c) implies (a), coherenceyields the conclusion that ba(ta) satisfies anonymity in beliefs for all ta.Without coherence, an admissible commutation of "almost all" quantifierswould yield the conclusion that ba(τ(ϕ, a)) satisfies anonymity in beliefs forΠa-almost all ϕ. As is evident from the proof, this weakening of statement(a) does not affect the validity of the implication that (a) implies (b).

Which version of the result one prefers, depends on which interpretationof the belief function ba(·) one considers appropriate. If one thinks of thebeliefs ba(ta), ta ∈ T, as the "real" representation of the agent’s expectations,and of the prior Πa as a derived concept, one is likely to prefer the versionof the result given here, which involves anonymity in beliefs for all ta. Incontrast, if one thinks of the prior Πa as the "real" representation of theagent’s expectations, and of the belief function ba(·) as a derived concept,one is likely to prefer the version of the result that involves anonymity inbeliefs for Πa-almost all τ(ϕ, a) and that does not require coherence.

29

4.4 Macro Belief Functions and Agent-Specific Priors

A belief function ba(·) that satisfies anonymity in beliefs is fully characterizedby the associated macro belief function b∗a(·) where b∗a(ta) = ba(ta) D(·)−1

for all ta. In Proposition 3.6 above, I had shown that, for any type ta ofagent a, any measure β onM(T ), and any rich Fubini extension (Ωa(ta)×A−a,Fa(ta)A−a, Pa(ta) α−a), there exists an Fa(ta)A−a-measurablemapping τ from Ωa(ta) × A−a to T such that the associated belief ba(ta)satisfies anonymity in beliefs and the macro belief b∗a(ta) is equal to β, i.e.once richness of the Fubini extension is imposed, there is no restriction onthe scope of macro uncertainty. In the following, I establish an analogousresult for macro belief functions and agent-specific priors. The constructionis more complicated than in Proposition 3.6 because, as in Proposition 4.3,there is a need to ensure that the beliefs of different types are coherent andmeasurable.

Coherence of Macro Belief Functions The macro belief function b∗a(·)is said to be coherent if, for any ta and ta in T, the probability measuresb∗a(ta) and b

∗a(ta) are mutually absolutely continuous, i.e., any Borel

set B ⊂ M(T ) satisfies b∗a(B|ta) > 0 if and only if it also satisfiesb∗a(B|ta) > 0.

Measurability of Macro Belief Functions The macro belief function b∗a(·)is said to be measurable if, for any Borel set B ⊂ M(T ), the set oftypes with macro beliefs in B, i.e. the set ta ∈ T |b∗a(ta) ∈ B, is itselfa Borel subset of T.

Given that, for any B ⊂ M(T ), the equation b∗a(ta) = ba(ta) D(·)−1

impliesb∗a(B|ta) = ba(ta′a′∈A−a |D(ta′a′∈A−a) ∈ B|ta),

one easily verifies that, if the belief function ba(·) is coherent, the associatedmacro belief function b∗a(·) is also coherent. Similarly, it is easy to verifythat, if ba(·) is measurable, the associated macro belief function b∗a(·) is alsomeasurable. As a partial converse to these observations, the following resultshows, that, starting from a coherent and measurable function β(·) from TtoM(M(T )), one can always find a coherent and measurable belief functionba(·) that has β(·) as its macro belief function.

Proposition 4.6 Let β(·) be any coherent and measurable function from Tto M(M(T )). Let (Φ, Φ, Πa) be a complete probability space. If the Fubiniextension (Φ×A, ΦA, Πaα) is rich, there exists a mapping τ : Φ×A→ T

30

that is measurable with respect to ΦA and a coherent and measurable belieffunction ba(·) such that ba(τ(·, a)) is a regular conditional distribution forτ(·, a′)a′∈A−a given τ(·, a), and, moreover, for all ta ∈ T, ba(ta) satisfiesanonymity in beliefs and the associated macro belief b∗a(ta) coincides withβ(ta).

Proof. The argument is similar to the one in the proof of Proposition3.6, with (Ωa(ta),Fa(ta), Pa(ta)) replaced by (Φ, Φ, Πa). Given the functionβ(·), let Ψβ ∈ M(T ×M(T )) be such that, for some (arbitrary) marginaldistribution ΨT

β on T, one has

Ψβ(Bt ×Bδ) =

∫Bt

β(Bδ|ta)dΨTβ (ta) (4.21)

for all Borel sets Bt ⊂ T and Bδ ⊂ M(T ). By standard arguments, thereexists a pair (τβ(·, a), δβ(·)) of random variables on (Φ, Φ, Πa), taking valuesin T ×M(T ), such that

Πa (τβ(·, a), δβ(·))−1 = Ψβ, (4.22)

i.e. the probability distribution of (τβ(·, a), δβ(·)) is Ψβ.As in the proof of Proposition 3.6, let f be a measurable function from

M(T )× [0, 1] into T such that for any δ ∈M(T ),

` f(δ, ·)−1 = δ (4.23)

where ` is the uniform distribution on [0, 1]. Given this function f and therandom pair (τβ(·, a), δβ(·)), define a mapping τ(·, ·) from Ωa(ta) × A intoT by setting

τ(·, a) = τβ(·, a) (4.24)

and

τ(·, a′) = f(δβ(·), h(·, a′)) for a′ ∈ A−a, (4.25)

where h is the function given by the richness of the Fubini extension (Φ ×A, ΦA, Πaα). Given that τ(·, ·) is measurable with respect to ΦA−a,one easily verifies that τ(·, ·) is measurable with respect to ΦA, as claimedin the proposition.

For any ta ∈ T, set ba(ta) =∫M(T ) ba(δ)dβ(δ|ta)) where ba is the func-

tion given by Remark 3.7. In the remainder of the proof, I will show thatthe function ba satisfies the claims made in the proposition. I first show

31

that ba(τ(·, a)) is a regular conditional distribution for τ(·, a′)a′∈A−a givenτ(·, a).

Let Cδ be the sub-σ-algebra of Φ that is generated by the random variableδβ(·). BecauseM(T ) is a complete separable metric space, Cδ is countablygenerated. Because the random variables h(·, a′), a′ ∈ A−a, are essentiallypairwise independent, Proposition 3 in Hammond and Sun (2006) impliesthat they are also essentially pairwise conditionally independent given Cδ.As in Remark 1 of Hammond and Sun (2008), it follows that the randompairs (δ(·), h(·, a′)), a′ ∈ A−a, are also essentially pairwise conditionally inde-pendent given Cδ and so are the random variables τ(·, a′) = f(δ(·), h(·, a′)),a′ ∈ A−a. By Proposition 3 in Qiao et al. (2016), these random variablesare also essentially pairwise exchangeable.

Because, for α−a-almost every a′ ∈ A−a, the random variable h(·, a′)has the uniform distribution `, (4.23) and (4.25) imply that, for α−a-almostevery a′ ∈ A−a, conditional on the event δβ(·) = δ, the probability distribu-tion of τa(·, a′) is almost surely equal to δ. For α−a-almost every a′ ∈ A−a,therefore the function δβ(·) is a regular conditional distribution for the ran-dom variable τa(·, a′) given Cδ.

Next, let Cτδ be the sub-σ-algebra of Φ that is generated by the pair(τβ(·, a), δβ(·)). Then Cτ δ is also countably generated, and Cδ ⊂ Cτδ. Byanother application of Proposition 3 in Hammond and Sun (2006), it fol-lows that the random variables τ(·, a′) = f(δ(·), h(·, a′)), a′ ∈ A−a, are alsoessentially pairwise conditionally independent given Cτδ. Moreover, by theargument in the proof of Proposition 3 of Hammond and Sun (2006), foralmost every a′ ∈ A−a, the function δβ(·) is also a regular conditional distri-bution for the random variable τa(·, a′) given Cτδ. δ is a regular conditionaldistribution for τ(·, a′) given Cτδ.

32

For any measurable function g on T ×Ra, we therefore obtain∫Φg(τ(ϕ, a), τ(ϕ, a′)a′∈A−a)dΠa(ϕ)

=

∫Φ

∫Ra

g(τ(ϕ, a), ta′a′∈A−a)dba(ta′a′∈A−a |δβ(ϕ))dΠa(ϕ)

=

∫T×M(T )

∫Ra

g(ta, ta′a′∈A−a)dba(ta′a′∈A−a |δ)dΨβ(ta, δ)

=

∫T

∫M(T )

∫Ra

g(ta, ta′a′∈A−a)dba(ta′a′∈A−a |δ)dβ(δ|ta)dΨT (ta)

=

∫Φ

∫M(T )

∫Ra

g(τ(ϕ, a), ta′a′∈A−a)dba(ta′a′∈A−a |δ)dβ(δ|τ(ϕ, a))dΠa(ϕ)

=

∫Φ

∫M(T )

∫Ra

g(τ(ϕ, a), ta′a′∈A−a)dba(ta′a′∈A−a |τ(ϕ, a))dΠa(ϕ),

which proves that ba(τ(·, a)) is in fact a regular conditional distribution forτ(·, a′)a′∈A−a given τ(·, a).

Turning to the other claims in the proposition, coherence and measura-bility of ba follow from the coherence and measurability of β. Because, con-ditionally on Cτδ, the random variables τa(·, a′|ta), a′ ∈ A−a, are essentiallypairwise independent, Corollary 2 of Qiao et al. (2016) implies that

D(τ(ϕ, a′)a′∈A−a) = δβ(ϕ),

Πa-almost surely, and therefore that the macro belief b∗a(ta) coincides withβ(ta) for all ta. Finally, anonymity in beliefs follows from the essential pair-wise exchangeability of the random variables τ(·, a′), a′ ∈ A−a.

5 Common Priors

5.1 Common Priors with Anonymity in Beliefs

The incomplete-information model T,Θ, θa, baa∈A is said to admit a com-mon prior if there exist a complete probability space (Φ, Φ, Π) and a map-ping τ : Φ×A→ T that is measurable with respect to a Fubini extension ofthe σ-algebra Φ⊗A such that for α-almost all a ∈ A, ba(τ(·, ϕ)) is a regularconditional distribution for τ(·, a′)a′∈A−a given τ(·, ϕ).

In the following, I discuss the existence of a common prior in the incomplete-information model T,Θ, θa, baa∈A. As is well known, even in models with

33

finitely many agents, the existence of a common prior cannot be taken forgranted.15 The conditions on the belief system under which a common priorexists in models with finitely many agents are very restrictive, the more sothe more agents there are.16

Having nothing to say on the general case, I focus on the case wherethe beliefs ba(τ(ϕ, a)) satisfy the condition of anonymity in beliefs. Thefollowing result is an immediate consequence of Proposition 4.5.

Proposition 5.1 Assume that the belief functions ba(·), a ∈ A, are coher-ent and measurable. Let (Φ, Φ, Π) and τ : Φ × A → T be such that theprobability space (Φ, Φ, Π) is complete, the mapping τ is measurable with re-spect to a Fubini extension of the σ-algebra Φ⊗A, and ba(τ(·, a)) is a regularconditional distribution for τ(·, a′)a′∈A−a given τ(·, a). Then the followingstatements are equivalent:

(a) For α-almost a ∈ A and all ϕ ∈ Φ, ba(τ(ϕ, a)) satisfies anonymityin beliefs.

(b) Under the measure Π, the random variables τ(·, a′), a′ ∈ A, are es-sentially pairwise exchangeable, i.e., there exists a probability measure pa(·)on T 2 such that, for α-almost all a1 ∈ A,

Π(τ(·, a1)−1(B1) ∩ τ(·, a2)−1(B2)) = pa(B1 ×B2) = pa(B2 ×B1) (5.1)

for α-almost all a2 ∈ A and all Borel sets B1, B2 ⊂ T.(c) If D ⊂ Φ is the Borel σ-algebra that is generated be the mapping ϕ→

D(τ(ϕ, a′)a′∈A), then conditionally on D, the random variables τ(·, a′),a′ ∈ A, are essentially pairwise conditionally independent and identicallydistributed. Moreover, the sample distribution D(τ(ϕ, a′)a′∈A) of types inthe population is almost surely equal to the common conditonal distributionof types in the population, i..e., for α-almost all a ∈ A, D(τ(·, a′)a′∈A) isa regular conditional distribution for τ(·, a) given D.

Proposition 5.1 provides a mathematical foundation for the applied-theory work mentioned in the introduction where uncertainty is assumedto have an aggregate component and an agent-specific component, withsymmetry and a law of large numbers holding for the latter. Whereas thisdecomposition of uncertainty is usually introduced ad hoc in order to sim-plify the analysis, Proposition 5.1 shows that it is actually implied by the

15Harsanyi (1967/68), Samet (1998 a, b), Feinberg (2000), Rodriguez-Neto (2010), Hell-man and Samet (2012), Hellwig (2012).16Hellman and Samet (2012).

34

assumption of anonymity in beliefs. In settings in which agents do notcare about the names of the other agents but only about the distributionof the other agents’types, uncertainty can always be decomposed into anaggregate component and an agent-specific component. The aggregate com-ponent concerns the cross-section distribution of types, the agent-specificcomponent concerns the different agents’ individual types. Conditionallyon the aggregate component, the agent-specific components are essentially(pairwise) independent and identically distributed, the conditional distrib-ution of any one agent’s type is equal to the cross-section distribution, andan exact (conditional) law of large numbers holds.

5.2 Common Priors and Macro Belief Functions

Whereas the belief functions ba in Proposition 5.1 differ across agents, dueto their having different domains and ranges, the exchangeability of therandom variables τ(·, a′), a′ ∈ A, ensures that the associated macro belieffunctions are the same for all agents. Thus, for any incomplete-informationmodel that admits a common prior and has beliefs satisfying the conditionof anonymity in beliefs, there exists a function β : T → M(M(T )) suchthat, for any agent a, β(τ(·, a)) is a regular conditional distribution forD(τ(·, a′)a′∈A) given τ(·, a). Moreover, of the belief functions ba, a ∈ A,are coherent and measurable, β is also coherent and measurable.

Suppose we take the macro belief function β as given. If β is coherent andmeasurable, can we infer the existence of a set of belief functions ba, a ∈ A,satisfying anonymity in beliefs such that the incomplete-information modelT,Θ, θa, baa∈A admits a common prior and β is the associated macro belieffunction? The answer to this question turns out to be negative. This findingcontrasts with Proposition 4.6 showing that for any coherent and measurablemacro belief function β and any a ∈ A, there exist an agent-specific priorand a coherent and measurable belief function ba satisfying anonymity inbeliefs such that, under the given prior, ba(ta) is a conditional probabilitydistribution for τ(·, a′)a′∈A given that the agent’s own type is ta.

In the proof of Proposition 4.6, the random variable τ(·, a) for the agent’sown type was specified without regard to the other agents’beliefs; the mar-ginal distribution ΨT

β of this random variable was arbitrary. In a model witha common prior, τ(·, a) cannot be specified arbitrarily but must be consis-tent with the other agents’beliefs. The following result gives a necessaryand suffi cient condition for this requirement to be satisfied.

Proposition 5.2 Let (Φ, Φ, Π) be a complete probability space and let (Φ×

35

A, Φ A, Π α) be a rich Fubini extension. For any coherent measurablefunction β : T →M(M(T )), the following statements are equivalent.

(a) There exists a mapping τ : Φ×A→ T that is measurable with respectto ΦA and coherent and measurable belief functions ba(·), a ∈ A, such that,for α-almost every a ∈ A, ba(τ(·, a)) is a regular conditional distribution forτ(·, a′)a′∈A−a given τ(·, a), and, moreover, for all ta ∈ T, ba(ta) satisfiesanonymity in beliefs and the associated macro belief b∗a(ta) coincides withβ(ta).

(b) There exist distributions ΨTβ on T and Ψ

M(T )β on M(T ) such that,

for all measurable sets Bt ⊂ T and Bδ ⊂M(T ),∫Bδ

δ(Bt)dΨM(T )β (δ) =

∫Bt

β(Bδ|t)dΨTβ (t). (5.2)

Proof. Suppose that statement (a) is true. Let Ψβ be the joint distributionof the pair (τ(·, a), D(τ(·, a′)a′∈A−a)). Let a be such that, for all ta ∈ T,ba(ta) satisfies anonymity in beliefs and the associated macro belief b∗a(ta)satisfies b∗a(ta) = β(ta). If ΨT

β ∈ M(T ) and ΨM(T )β ∈ M(M(T )) are the

associated marginal distributions on T andM(T ), then, for all measurablesets Bt ⊂ T and Bδ ⊂M(T ),

Ψβ(Bt ×Bδ) =

∫Bt

β(Bδ|t)dΨTβ (t) (5.3)

and

Ψβ(Bt ×Bδ) =

∫Bδ

δ(Bt)dΨM(T )β (δ). (5.4)

Equation (5.3) must hold because, by the definition of the macro belieffunction b∗a and the coincidence of b

∗a and β, β(τ(·, a)) is a regular condi-

tional distribution for D(τ(·, a′)a′∈A−a) given τ(·, a) and, moreover, bythe atomlessness of α, D(τ(·, a′)a′∈A−a) = D(τ(·, a′)a′∈A). Equation(5.4) must hold because, by the assumption that, for all ta ∈ T, ba(ta)satisfies anonymity in beliefs, statement (c) in Proposition 4.5 implies thatD(τ(·, a′)a′∈A−a) = D(τ(·, a′)a′∈A) is a regular conditional distributionfor τ(·, a) given D(τ(·, a′)a′∈A−a). Upon combining (5.3) and (5.4), oneobtains (5.2), which proves statement (b).

Conversely, suppose that statement (b) is true. Fix some a ∈ A. Usethe construction in the proof of Proposition 4.6 with the given ΨT

β to obtainthe desired τ(·, ·) and ba(·) (for this particular agent). The joint distrib-ution of the pair (τ(·, a), D(τ(·, a′)a′∈A−a)) is then given by (5.3). ByProposition 4.5, for α-almost all a′′ ∈ A−a, D(τ(·, a′)a′∈A−a) is a regular

36

conditional distribution for τ(·, a′′) given D(τ(·, a′)a′∈A−a). By (5.4), itfollows that, for α-almost all a′′ ∈ A−a, Ψβ is also the joint distribution ofthe pair (τ(·, a′′), D(τ(·, a′)a′∈A−a)). By another application of (5.3), there-fore, β(τ(·, a′′)) is a regular conditional distribution for D(τ(·, a′)a′∈A) =D(τ(·, a′)a′∈A−a′′ ) given τ(·, a′′). Upon setting

ba′′(ta′′) =

∫M(T )

ba′′(δ)dβ(δ|ta′′)),

with ba′′ given by Remark 3.7, one obtains the desired belief function ba′′ foragent a′′. The same arguments as in the proof of Proposition 4.5 show thatthis belief function has the properties postulated in statement (a).

There are two ways to think about the joint distribution Ψβ of the pair(τ(·, a), D(τ(·, a′)a′∈A−a)). One way is to think about τ(·, a) as given andto derive Ψβ from the distribution ΨT

β of τ(·, a) and the macro belief functionβ. Another way is to think about D(τ(·, a′)a′∈A−a) = D(τ(·, a′)a′∈A) as

given and to derive Ψβ from the distribution ΨM(T )β of D(τ(·, a′)a′∈A)

and the fact that, conditionally on D(τ(·, a′)a′∈A), τ(·, a) is distributed asD(τ(·, a′)a′∈A). Statement (b) in Proposition 5.2 ensures that these twoapproaches are mutually consistent.

5.3 Existence of a Common Prior

The condition that Proposition 5.2 gives for the existence of a common priorin a model with a continuum of agents and anonymity in beliefs is formallyequivalent to a condition for the existence of a common prior in a certaintwo-player model. In this two-player model, player 1 has the type space Tand player 2 the type spaceM(T ). Beliefs are given by the functions t→ β(t)and δ → δ. For each t ∈ T, the probabilistic beliefs of player 1 about thetype of player 2 are given by β(t) ∈ M(M(T )); for each δ ∈ M(T ), theprobabilistic beliefs of player 2 about the type of player 1 are given by δ.For given β, a measure Ψβ ∈ M(T ×M(T )) that satisfies statement (b)in Proposition 5.2 exists if and if Ψβ is a common prior for the specifiedtwo-player model with belief functions t→ β(t) and δ → δ.

Given this equivalence, I use arguments from the analysis of two-playergames to give a condition on the macro belief function β under which thisfunction is compatible with the existence of a common prior. I begin witha lemma showing that coherent macro beliefs have densities and that thesedensities are positive.

37

Lemma 5.3 If the macro belief function β(·) is coherent, there exists ameasurable functions f : M(T ) × T × T → R+ such that, for any t and t′

in T and any measurable set Bδ ⊂M(T ),

β(Bδ|t′) =

∫Bδ

f(δ, t′, t) β(dδ|t). (5.5)

The function f satisfies the equation

f(δ, t′′, t) = f(δ, t′′, t′) · f(δ, t′, t) (5.6)

for β(t)-almost all δ ∈M(T ), for all t, t′, t′′ in T. Moreover,

f(δ, t, t′) > 0 (5.7)

for β(t′)-almost all δ ∈ D, for all t, t′ ∈ T.

Proof. The first statement of the lemma follows from strong coherence andthe Radon-Nikodym Theorem. To prove the second statement, it suffi cesto note that, for any measurable set Bδ ⊂ M(T ) and any t, t′, t′′, the firststatement implies

β(Bδ|t′′) =

∫Bδ

f(δ, t′′, t′) dβ(δ|t′) =

∫Bδ

f(δ, t′′, t′) · f(δ, t′, t) dβ(δ|t),

so that the product f(·, t′′, t′)·f(·, t′, t) is a Radon-Nikodym derivative of themeasure β(t′′) with respect to the measure β(t). The statement follows be-cause the Radon-Nikodym derivative of one measure with respect to anotheris unique up to a set of measure zero.

To prove the last statement of the lemma, fix t′ ∈ T and δ′ ∈ D. Uponsetting t′′ = t in (5.6), one finds that f(δ, t, t′) · f(δ, t′, t) = 1 for β(t)-almostall δ ∈ M(T ), for all t and t′ in T. Therefore f(δ, t, t′) > 0 for β(t)-almostall δ ∈M(T ), for all t and t′ in T. The last statement of the lemma followsbecause the measures β(t) and β(t′) are mutually absolutely continuous.

I next introduce a property ensuring that cross-section type distributionsin the support of the macro beliefs β(t), t ∈ T, also have positive densities.For lack of a better term, I call this property strong coherence.

Strong Coherence of Macro Belief Functions Amacro belief functionβ : T →M(M(T )) is said to be strongly coherent if it is coherent andif there exists a set D ⊂M(T ) such that β(D|t) = 1 for all t ∈ T andthe measures δ ∈ D are mutually absolutely continuous.

38

Lemma 5.4 If the macro belief function β(·) is strongly coherent, thereexists a measurable function g : T ×D ×D → R+ such that, for any δ andδ′ in the specified set D and any measurable set Bt ⊂ T,

δ′(Bt) =

∫Bt

g(t, δ′, δ) δ(dt). (5.8)

The function g satisfies the equation

g(t, δ′′, δ) = g(t, δ′′, δ′) · g(t, δ′, δ) (5.9)

for δ-almost all t ∈ T, for all δ, δ′, δ′′ in D. Moreover,

g(t, δ, δ′) > 0 (5.10)

for δ′-almost all t ∈ T, for all δ all δ′ ∈ D.

The proof of Lemma 5.4 is step for step the same as the proof of Lemma5.3 and is left to the reader.

Proposition 5.5 A strongly coherent macro belief function β : T →M(M(T ))admits a common prior if and only if the associated density functions f andg satisfy the equation

g(t1, δ1, δ0) · f(δ1, t2, t0) · g(t2, δ2, δ0) · f(δ2, t1, t0)

= f(δ1, t1, t0) · g(t1, δ2, δ0) · f(δ2, t2, t0) · g(t2, δ1, δ0) (5.11)

for any t0 ∈ T and δ0 ∈ D, for β(t0)-almost all δ1 and δ2 in D and δ0-almostall t1 and t2 in T .

Proof. If β admits a common prior, then statement (b) in Proposition 5.2holds. Given the marginal distributions ΨT

β and ΨM(T )β specified let there,

let Ψβ be the distribution onM(T ×M(T )) that is given by (5.3) and (5.4).

I note that the measure ΨM(T )β and any one of the measures β(t0), t0 ∈ T,

are mutually absolutely continuous: For Bt = T and any t0 ∈ T, (5.3) andLemma 5.3 imply

ΨM(T )β (Bδ) =

∫Tβ(Bδ|t) dΨT

β (t)

=

∫T

∫Bδ

f(δ, t, t0) dβ(δ|t0) dΨTβ (t)

=

∫Bδ

∫Tf(δ, t, t0) dΨT

β (t) dβ(δ|t0). (5.12)

39

Absolute continuity of ΨM(T )β with respect to β(t0) is immediate. Absolute

continuity of β(t0) with respect to ΨM(T )β follows from the strict positivity

of f(δ, t, t0).I next claim that the measure ΨT

β and any measure δ0 ∈ D are mutuallyabsolutely continuous. To see this, note that, for Bδ = D, equation (5.4)implies

ΨTβ (Bt) =

∫Dδ(Bt) dΨ

M(T )β (δ)

=

∫D

∫Bt

g(t, δ, δ0) dδ0(t) dΨM(T )β (δ)

=

∫Bt

∫Dg(t, δ, δ0) dΨ

M(T )β (δ) dδ0(t) (5.13)

for any δ0 ∈ D. Hence ΨTβ is absolutely continuous with respect to δ0. By

the strict positivity of g(t, δ, δ0), (5.13) also implies that δ0 is absolutelycontinuous with respect to ΨT

β .Equations (5.13) and (5.12) also imply that the Radon-Nikodym deriv-

atives of ΨTβ and Ψ

M(T )β with respect to δ0 and β(t0) are given as

ψt(·|δ0) :=

∫Dg(·, δ) dΨ

M(T )β (δ) (5.14)

and

ψδ(·|t0) :=

∫Tf(·, t, t0) dΨT

β (t). (5.15)

Thus, equations (5.3) and (5.4) can be rewritten in the form

Ψβ(Bt ×Bδ) =

∫Bδ

∫Bt

g(t, δ) · ψδ(δ|t0) dδ0(t) dβ(δ|t0)

and

Ψβ(Bt ×Bδ) =

∫Bt

∫Bδ

f(δ, t, t0) · ψt(t|δ0) dβ(δ|t0) dδ0(t).

One easily sees that Ψβ is absolutely continuous with respect to the productmeasure β(t0)× δ0, with a density ψ(·, ·|t0, δ0) satisfying

ψ(t, δ|t0, δ0) = g(t, δ, δ0) · ψδ(δ|t0) = f(δ, t, t0) · ψt(t|δ0) (5.16)

for δ0-almost all t ∈ T and β(t0)-almost all δ ∈ D.

40

Consider the second equation in (5.16) with t and δ replaced by differentconstellations of t1, t2 ∈ T and δ1, δ2 ∈ D. This yields the equation

g(t1, δ1) · ψδ(δ1, t0) · f(δ1, t2, t0) · ψt(t2)

·g(t2, δ2) · ψδ(δ2, t0) · f(δ2, t1, t0) · ψt(t1)

= (5.17)

f(δ1, t1, t0) · ψt(t1) · g(t1, δ2) · ψδ(δ2, t0)

·f(δ2, t2, t0) · ψt(t2) · g(t2, δ1) · ψδ(δ1, t0).

Because the measures ΨTβ and δ0, as well as the measures Ψ

M(T )β and β(t0),

are mutually absolutely continuous, the same argument as in the proof ofLemma 5.3 implies that

ψt(t, δ0) > 0 for δ0-almost all t

andψδ(δ, t0) > 0 for β(t0)-almost all δ.

For δ0-almost all t1 and t2 in T and β(t0)-almost all δ1 and δ2 inD, therefore,the terms ψδ(δ1, t0), ψt(t2), ψδ(δ2, t0), and ψt(t1) in (5.17) can be cancelled,which yields (5.11). Because the choice of t0 and δ0 was arbitrary, theconclusion holds for all t0.

To prove the converse, suppose that, for some t0 ∈ T and δ0 ∈ D,equation (5.11) is satisfied for δ0-almost all t1 and t2 in T and β(t0)-almostall δ1 and δ2 in D. By Lemma 5.3, there exist some t1 and δ1 so that

f(δ1, t1, t0) > 0, g(t1, δ1) > 0,

andg(t1, δ) > 0 for β(t0)-almost all δ.

For any t and δ, define

λ(t0, δ0) :=

[∫D

∫T

f(δ, t1, t0)

f(δ1, t1, t0)· g(t, δ, δ0)

g(t1, δ, δ0)dδ0(t)dβ(δ|t0)

]−1

, (5.18)

ψ(t, δ|t0, δ0) := λ(t0, δ0) · f(δ, t1, t0)

f(δ1, t1, t0)· g(t, δ, δ0)

g(t1, δ, δ0)(5.19)

if g(t1, δ, δ0) > 0, andψ(t, δ|t0, δ0) ::= 0

41

if g(t1, δ, δ0) = 0. Further define a measure Ψβ ∈M(T ×D) by setting

Ψβ(Bt ×Bδ) :=

∫Bδ

∫Bt

ψ(t, δ|t0, δ0) dδ0(t)dβ(δ|t0) (5.20)

for any measurable Bt ⊂ T and Bδ ⊂ D. Then, by construction,

Ψβ(Bt ×Bδ) =

∫Bδ

∫Bt

λ · f(δ, t1, t0)

f(δ1, t1, t0)· g(t, δ, δ0)

g(t1, δ, δ0)dδ0(t)dβ(δ|t0)

=

∫Bδ

∫Bt

g(t, δ, δ0) dδ0(t) · dΨM(T )β (δ)

=

∫Bδ

∫Bt

dδ(t) · dΨM(T )β (δ), (5.21)

where ΨM(T )β ∈M(M(T )) is given by the formula

ΨM(T )β (Bδ) = Ψβ(T ×Bδ) =

∫Bδ

λ · f(δ, t1, t0)

f(δ1, t1, t0)· 1

g(t1, δ, δ0)dβ(δ|t0).

Equation (5.21) shows that Ψβ satisfies (5.4).To verify that Ψβ also satisfies (5.3), observe that, by Lemma 5.3 and

(5.19), we also have

ψ(t, δ|t0, δ0) = λ · f(δ, t, t0)

f(δ1, t, t0)· g(t, δ1, δ0)

g(t1, δ1, δ0)(5.22)

for δ0 × β(t0)-almost all (t, δ). Therefore, (5.20) can be rewritten as

Ψβ(Bt ×Bδ) =

∫Bt

∫Bδ

λ · f(δ, t, t0)

f(δ1, t, t0)· g(t, δ1, δ0)

g(t1, δ1, δ0)dδ0(t)dβ(δ|t0)

=

∫Bt

∫Bδ

f(δ, t, t0) dβ(δ|t0) · dΨTβ (t)

=

∫Bt

∫Bδ

dβ(δ|t) dΨTβ (t), (5.23)

where ΨTβ ∈M(T ) is given by the formula

ΨTβ (Bt) = Ψβ(Bt ×D) =

∫Bt

λ · 1

f(δ1, t, t0)· g(t, δ1, δ0)

g(t1, δ1, δ0)dδ0(t).

Equation (5.23) shows that Ψβ also satisfies (5.3). Thus, the macro belieffunction β satisfies statement (b) in Proposition 5.2.

42

The consistency condition (5.11) is well known from Harsanyi (1967/68).Whereas most of the literature discusses this condition in terms of a two-player game with finitely many states, Proposition 4.5 shows that it also mat-ters for the existence of a common prior in a large eocnomy with anonymity,with a potentially uncountable set of states. The underlying logic is thesame: If distribution Ψβ has a density ψ(·, ·|t0, δ0) with respect to the mea-

sure δ0 × β(t0), the ratio ψ(t′,δ′|t0,δ0)ψ(t,δ|t0,δ0) can be computed from the equation

ψ(t′, δ′|t0, δ0)

ψ(t, δ|t0, δ0)=ψ(t′, δ|t0, δ0)

ψ(t, δ|t0, δ0)· ψ(t′, δ′|t0, δ0)

ψ(t′, δ|t0, δ0)(5.24)

or from the equation

ψ(t′, δ′|t0, δ0)

ψ(t, δ|t0, δ0)=ψ(t, δ′|t0, δ0)

ψ(t, δ|t0, δ0)· ψ(t′, δ′|t0, δ0)

ψ(t, δ′|t0, δ0), (5.25)

and each time the result must be the same. If Ψβ satisfies (5.3) and (5.4),the density ψ(·, ·|t0, δ0) satisfies (5.16), which yields

ψ(t′, δ|t0, δ0)

ψ(t, δ|t0, δ0)=g(t′, δ, δ0)

g(t, δ, δ0)(5.26)

andψ(t, δ′|t0, δ0)

ψ(t, δ|t0, δ0)=f(δ′, t, t0)

f(δ, t, t0)(5.27)

for any t, t′ ∈ T and any δ, δ′ ∈ D such that ψ(t, δ|t0, δ0) > 0 (and thereforeg(t, δ, δ0) > 0 and f(δ, t, t0) > 0). When applied to (5.24), (5.26) and (5.27)yield

ψ(t′, δ′|t0, δ0)

ψ(t, δ|t0, δ0)=f(δ′, t, t0)

f(δ, t, t0)· g(t′, δ′, δ0)

g(t, δ′, δ0), (5.28)

when applied to (5.25),

ψ(t′, δ′|t0, δ0)

ψ(t, δ|t0, δ0)=g(t′, δ, δ0)

g(t, δ, δ0)· f(δ′, t′, t0)

f(δ, t′, t0). (5.29)

For these two expressions to be compatible, one needs equation (5.11).Notice that, according to (5.28) and (5.29), the ratio ψ(t′,δ′|t0,δ0)

ψ(t,δ′|t0,δ0)is uniquely

determined by the density functions f and g. Because the integral of ψ(·, ·|t0, δ0)with respect to the measure δ0×β(t0) must be equal to one, it follows that,up to modifications on a set of β(t0)× δ0-measure zero, the density functionψ(·, ·|t0, δ0) and the measure Ψβ are uniquely determined by the functionsf and g.

43

Remark 5.6 The common prior that is obtained if the condition in Propo-sition 5.2 holds is unique.

Remark 5.6 implies, in particular, that Ψβ does not depend on the pair(t1, δ1) that serves as a starting point for the construction in (5.18) - (5.20).The following remark shows that Ψβ also does not depend on the choice ofthe pair (t0, δ0) in equation (5.11) and the construction in (5.18) - (5.20).

Remark 5.7 The common prior that is obtained if the condition in Propo-sition 5.2 holds does not depend on the particular pair (t0, δ0) that are usedin the construction.

Proof. I first note that the validity of (5.11) does not depend on the cho-sen pair (t0, δ0). To see this, multiply both sides of (5.11) by the productg(t1, δ0, δ

′0)·f(δ1, t0, t

′0)·g(t2, δ0, δ

′0)·f(δ2, t0, t

′0), for some t′0 ∈ T and δ′0 ∈ D,

and simplify the resulting equation using (5.6) and (5.9). The result is aversion of (5.11)with (t0, δ0) replaced by (t′0, δ

′0).

To prove that Ψβ is unchanged if, in the construction (5.18) - (5.20),(t0, δ0) replaced by (t′0, δ

′0), I note that, by Lemmas 5.3 and 5.4, the right-

hand side of (5.20) can be written as∫Bδ

∫Bt

f(δ, t1, t0) · f(δ, t0, t′0)

f(δ1, t1, t0)· g(t, δ, δ0) · g(t, δ0, δ

′0)

g(t1, δ, δ0)dδ′0(t)dβ(δ|t′0)).

By another application of Lemmas 5.3 and 5.4, therefore, (5.20) can berewritten as

Φ(Bt ×Bδ) =

∫Bδ

∫Bt

λ′ · f(δ, t1, t′0)

f(δ1, t1, t′0)· g(t, δ, δ′0)

g(t1, δ, δ′0)dδ′0(t)dβ(δ|t′0))

=λ′

λ(t′0, δ′0)

∫Bδ

∫Bt

ψ(t, δ|t′0, δ′0) dδ′0(t)dβ(δ|t′0)), (5.30)

where λ′ := λ(t0, δ0) · f(δ1, t0, t′0) · g(t1, δ0, δ

′0) and λ(t′0, δ

′0) and ψ(t, δ|t′0, δ′0)

are given by (5.18) and (5.19) with (t0, δ0) replaced by (t′0, δ′0). For Bt = T

andBδ = D, we have Φ(Bt×Bδ) = 1 and∫Bδ

∫Btψ(t, δ, t′0, δ

′0) δ′0(dt)β(dδ|t′0)) =

1. Therefore, λ′ = λ(t′0, δ′0), and (5.30) can be rewritten as

Φ(Bt ×Bδ) =

∫Bδ

∫Bt

ψ(t, δ|t′0, δ′0) δ′0(dt)β(dδ|t′0))

for all Bt and Bδ, which is just (5.20) with (t0, δ0) replaced by (t′0, δ′0).

44

Whereas the Harsanyi consistency condition is usually discussed as anecessary condition for the existence of a common prior, Proposition 4.5shows that, under the given conditions, it is also suffi cient. This resulthinges on the strict positivity of the densities f and g, which in turn isderived from the assumption that the macro belief function β is stronglycoherent.17

Strong coherence involves two properties, (i) coherence, i.e., mutual ab-solute continuity of the measures β(t), t ∈ T, and (ii) mutual absolute con-tinuity of the measures δ ∈ D, where β(D|t) = 1 for all t. Coherence of themeasures β(t), t ∈ T, is restrictive, but, as discussed in Section 4, this prop-erty plays a key role in establishing that a given belief function ba admitseven an agent-specific prior, let alone a common prior.

Given the coherence of the measures β(t), t ∈ T, the additional conditionof mutual absolute continuity of the measures δ ∈ D, where β(D|t) = 1 forall t, is close to being necessary for the existence of a common prior (as wellas being suffi cient, in combination with the Harsanyi consistency condition).

Remark 5.8 Suppose that a coherent macro belief function β : T →M(M(T ))admits a common prior Ψβ ∈ M(T ×M(T )), with marginal distributions

ΨTβ ,Ψ

M(T )β on T andM(T ). Then for every measurable set Bt ⊂ T, ΨT

β (Bt) =0 if and only if δ(Bt) = 0 for β(t0)-almost all δ ∈M(T ), for every t0 ∈ T.

Proof. Fix Bt ⊂ T. By (5.4),

ΨTβ (Bt) =

∫M(T )

δ(Bt)dΨM(T )β (δ), (5.31)

so ΨTβ (Bt) = 0 implies Ψ

M(T )β (Bδ(Bt)) = 1, where

Bδ(Bt) = δ ∈M(T )|δ(Bt) = 0.

By (5.3) and Lemma 5.3, we also have

ΨM(T )β (Bδ) =

∫Bδ

∫Tf(δ, t, t0) dΨT

β (t) dβ(δ|t0),

17This suffi ciency result parallels the finding in Hellwig (2013) that, for games withfinitely many players, if any element of any player’s information partition intersects anyelement of any other player’s information partition, then, for strictly positive belief systems,a common prior exists if the Harsanyi condition holds for all "cycles" of length four orless.

45

for every measurable set Bδ ⊂M(T ) and every t0 ∈ T. Since f(δ, t, t0) is al-most everywhere strictly positive, it follows that Ψ

M(T )β and β(t0) are mutu-

ally absolutely continuous, soΨM(T )β (Bδ(Bt)) = 1, i.e., ΨM(T )

β (M(T )\Bδ(Bt)) =0, if and only if β(M(T )\Bδ(Bt)) = 0, i.e., β(Bδ(Bt)|t0) = 1. The remarkfollows immediately.

In the condition that, for every Bt ⊂ T, ΨTβ (Bt) = 0 if and only if

δ(Bt) = 0 for β(t0)-almost all δ ∈M(T ), for every t0 ∈ T, the β(t0)-null setof measures δ for which δ(Bt) > 0 even as ΨT

β (Bt) = 0 might depend on Bt.Therefore this condition is slightly weaker than mutual absolute continuityof ΨT

β and δ, for β(t0)-almost all δ ∈ M(T ), which in turn would implystrong coherence.

If T was a finite set or if the measures β(t), t ∈ T, had finite supports,the condition given in Remark 5.8 would actually be suffi cient for mutualabsolute continuity ofΨT

β and δ, for β(t0)-almost all δ ∈M(T ), and thereforefor strong coherence of β. In this case, Proposition 4.5 could be reformulatedas a result about coherent macro belief functions, with strong coherenceappearing as a part of the necessary and suffi cient condition for the existenceof a common prior.

46

References

[1] Alesina, A., and G. Tabellini (1990), A Positive Theory of Fiscal Deficitsand Government Debt, Review of Economic Studies 87, 403-414.

[2] Angeletos, G.-M., and I. Werning (2006), Crises and Prices: Infor-mation Aggregation, Multiplicity, and Volatility, American EconomicReview 96 (5), 1720-1736.

[3] Bierbrauer, F.J., and M.F. Hellwig (2015), Public-GoodProvision in Large Economies, Preprint 2015-12, MaxPlanck Institute for Research on Collective Goods, Bonn,http://www.coll.mpg.de/pdf_dat/2015_12online.pdf.

[4] Bryant, J. (1980), A Model of Reserves, Bank Runs, and Deposit In-surance, Journal of Banking and Finance 4, 335-344.

[5] Diamond, D.W., and P.H. Dybvig (1983), Bank Runs, Deposit Insur-ance, and Liquidity, Journal of Political Economy 91, 401-419.

[6] Feinberg, Y. (2000), Characterizing common priors in the form of pos-teriors, Journal of Economic Theory 91, 127-179.

[7] Feldman, M., and C. Gilles (1985), An expository note on individualrisk without aggregate uncertainty, Journal of Economic Theory 35,26-32.

[8] Goldstein, I., and A. Pauzner (2005), Demand Deposit Contracts andthe Probability of a Bank Run, Journal of Finance 60 (3), 1293-1327.

[9] Grossman, S.J., and O.D. Hart (1980), Takeover Bids, the Free-RiderProblem, and the Theory of the Corporation, Bell Journal of Eco-nomics, 11(1), 42-64.

[10] Gul, F., H. Sonnenschein, and R. Wilson (1986) Foundations of Dy-namic Monopoly and the Coase Conjecture, Journal of Economic The-ory 39, 155-190

[11] Hammond, P.J., and Y. Sun (2003), Monte Carlo Simulation of Macro-economic Risk with a Continuum of Agents: the Symmetric Case, Eco-nomic Theory 21, 743-766

[12] Hammond, P.J., and Y. Sun (2006), The Essential Equivalence of Pair-wise and Mutual Conditional Independence, Probability Theory and Re-lated Fields 135, 415-427

47

[13] Hammond, P.J., and Y. Sun (2008), Monte Carlo Simulation of Macro-economic Risk with a Continuum of Agents: the General Case, Eco-nomic Theory 36, 303-325.

[14] Harsanyi, J.C. (1967/68), Games with Incomplete Information Playedby Bayesian Players I, II, III, Management Science 14, 159 - 182; 320 -334; 486-502.

[15] Hellman, Z., and D. Samet (2012), How Common are Common Priors?,Games and Economic Behavior 74, 517-525.

[16] Hellwig, C. (2002), Public Information, Private Information, and theMultiplicity of Equilibria in Coordination Games, Journal of EconomicTheory 107(2), 191-222.

[17] Hellwig, M.F. (2013), From Posteriors to Priors via Cycles: An Adden-dum, Economics Letters 118, 455-458.

[18] Hellwig, M.F. (2011), Incomplete-Information Models ofLarge Economies with Anonymity: Existence and Unique-ness of Common Priors, Preprint, No. 2011/8, MaxPlanck Institute for Research on Collective Goods, Bonn.http://www.coll.mpg.de/pdf_dat/2011_08online.pdf.

[19] Judd, K.L. (1985), The law of large numbers with a continuum of IIDrandom variables, Journal of Economic Theory 35, 19-25.

[20] Khan, M.A., and Y. Sun (1999), Non-cooperative games on hyperfiniteLoeb spaces, Journal of Mathematical Economics 31, 455—492.

[21] Kingman, J.F.C. (1978), Uses of Exchangeability, Annals of Probability6(2), 183-197.

[22] Kyle, A.S. (1985), Continuous Auctions and Insider Trading, Econo-metrica 53(6), 1315-1336.

[23] Kyle, A.S. (1989), Informed Speculation with Imperfect Competition,Review of Economic Studies 56 (3), Pages 317-355.

[24] Lindbeck, A., and J. Weibull (1987), Balanced-Budget Redistributionas the Outcome of Political Competition, Public Choice 52(3), 273-297.

[25] Mertens, J.-F., and S. Zamir (1985), Formulation of Bayesian analysisfor games with incomplete information, International Journal of GameTheory 14, 1-29.

48

[26] Morris, S. (1994), Trade with Heterogeneous Prior Beliefs and Asym-metric Information, Econometrica 62, 1326-1347.

[27] Morris, S., and H.S. Shin (1998), Unique Equilibrium in a Model of Self-Fulfilling Currency Attacks, American Economic Review 88(3), 587-597.

[28] Parthasarathy, K.R. (1967), Probability Measures on Metric Spaces,Academic Press, New York.

[29] Podczeck, K. (2010), On Existence of Rich Fubini Extensions, EconomicTheory 45, 1-22.

[30] Qiao, L., Y. Sun, and Z. Zhang (2016), Conditional Exact Law of LargeNumbers and Asymmetric Information Economies with Aggregate Un-certainty, Economic Theory 62(1-2), 43-64.

[31] Rochet, J.-C., and X. Vives (2004), Coordination Failures and theLender of the Last Resort, Journal of the European Economic Asso-ciation 2(6), 1116 - 1147.

[32] Rodrigues-Neto, J.A. (2009), From Posteriors to Priors via Cycles,Journal of Economic Theory 144, 876-883.

[33] Samet, D. (1998a), Iterative Expectations and Common Priors, Gamesand Economic Behavior 24, 131-141.

[34] Samet, D. (1998b), Common Priors and the Separation of Convex Sets,Games and Economic Behavior 24, 172-174.

[35] Sun, Y. (2006), The Exact Law of Large Numbers via Fubini Extensionand Characterization of Insurable Risks, Journal of Economic Theory126, 31-69.

[36] Sun, Y., and Y.Zhang (2009), Individual Risk and Lebesgue Extensionwithout Aggregate Uncertainty, Journal of Economic Theory 144, 432-442.

49